Nostr notes by MagicInternetMath

📖 Algebraic Closure An algebraic closure of $F$ is an ...

2026-09-03T21:53:37Z

📖 Algebraic Closure

An **algebraic closure** of $F$ is an algebraically closed field $\\overline{F}$ that is algebraic over $F$. It exists and is unique up to $F$-isomorphism. The **absolute Galois group** is $\\operatorname{Gal}(\\overline{F}/F)$.

From: gal-morandi
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📖 Mining Pool A mining pool aggregates many miners\ From: ...

2026-09-03T01:27:37Z

📖 Mining Pool

A mining pool aggregates many miners\

From: mining-pools
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📖 PPS / FPPS Pay-per-share: the pool pays a fixed price for ...

2026-09-02T23:33:41Z

📖 PPS / FPPS

Pay-per-share: the pool pays a fixed price for every share submitted, block or no block — the operator absorbs all luck variance. FPPS additionally pays the expected transaction-fee revenue per share.

From: mining-pools
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📐 Bonferroni Bound Testing $m$ cells at once, requiring ...

2026-09-02T21:18:21Z

📐 Bonferroni Bound

Testing $m$ cells at once, requiring per-test $p < \\alpha/m$ keeps the probability of even one false positive below $\\alpha$. For the era: $m = 1440$, $\\alpha = 0.05 \\Rightarrow p < 3.47 \\times 10^{-5}$.

Proof: Union bound: $P(\\text{any false positive}) \\leq \\sum_{i=1}^{m} P(\\text{test } i \\text{ false positive}) = m \\cdot (\\alpha/m) = \\alpha$.

From: mining-pools
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📖 Roots of Unity The roots of $x^n - 1$ form a cyclic group ...

2026-08-09T04:50:15Z

📖 Roots of Unity

The roots of $x^n - 1$ form a cyclic group under multiplication. A generator $\\epsilon$ (of order exactly $n$) is a primitive $n$-th root of unity.

From: gal-artin
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📖 Resolvent Equation The auxiliary equation of lower degree ...

2026-08-08T23:04:18Z

📖 Resolvent Equation

The auxiliary equation of lower degree that arises in the process of solving a polynomial equation is called its resolvent equation (or simply resolvent). For the quartic, the resolvent is a cubic. This pattern -- solving an equation by reducing it to a resolvent of lower degree -- is the fundamental idea that Lagrange would later analyze in full generality.

From: gal-edwards
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📐 Characteristic is Prime The characteristic of a field is ...

2026-08-08T22:02:24Z

📐 Characteristic is Prime

The characteristic of a field is either $0$ or a prime number $p$.

Proof: If $\\operatorname{char}(F) = n = ab$ with $1 < a, b < n$, then $(a \\cdot 1)(b \\cdot 1) = n \\cdot 1 = 0$. Since $F$ is a field (hence an integral domain), either $a \\cdot 1 = 0$ or $b \\cdot 1 = 0$, contradicting the minimality of $n$.

From: gal-weintraub
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📖 Linear Substitution Modulo q A permutation $\\sigma$ of ...

2026-08-08T17:09:21Z

📖 Linear Substitution Modulo q

A permutation $\\sigma$ of $\\{1, \\ldots, q\\}$ (with $q$ prime) is a linear substitution modulo $q$ if $\\sigma(i) \\equiv bi + c \\pmod{q}$ for integers $b \\not\\equiv 0$ and $c$. These form a group of order $q(q-1)$.

From: gal-artin
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📖 Galois Group The Galois group of the equation $f(x) = 0$ ...

2026-08-08T16:02:56Z

📖 Galois Group

The Galois group of the equation $f(x) = 0$ over the field $K$ is the group of substitutions of the roots $a, b, c, \\ldots$ presented by the table whose rows are $\\phi_a(t\

From: gal-edwards
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📐 Gram-Schmidt Procedure If $v_1, \\ldots, v_m$ is a linearly ...

2026-08-08T15:04:03Z

📐 Gram-Schmidt Procedure

If $v_1, \\ldots, v_m$ is a linearly independent list in $V$, then there exists an orthonormal list $e_1, \\ldots, e_m$ such that $\\operatorname{span}(v_1, \\ldots, v_j) = \\operatorname{span}(e_1, \\ldots, e_j)$ for each $j$.

From: linalg-axler
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🎮 Interactive: Determinant Visualizer Understand the ...

2026-08-08T14:18:17Z

🎮 Interactive: Determinant Visualizer

Understand the determinant as signed area/volume. See why det(AB) = det(A)det(B) and when matrices are invertible.

From: Linear Algebra
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📐 Fundamental Theorem of Galois Theory Let $E/K$ be a finite ...

2026-08-08T11:06:02Z

📐 Fundamental Theorem of Galois Theory

Let $E/K$ be a finite Galois extension with $G = \\mathrm{Gal}(E/K)$. There is an inclusion-reversing bijection between the set of intermediate fields $K \\subseteq F \\subseteq E$ and the set of subgroups $H \\leq G$, given by $F \\mapsto \\mathrm{Gal}(E/F)$ and $H \\mapsto E^H$. Moreover, $[E:F] = |\\mathrm{Gal}(E/F)|$ and $F/K$ is normal iff $\\mathrm{Gal}(E/F) \\trianglelefteq G$.

From: gal-jacobson
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📖 Affine Algebraic Variety An affine algebraic variety ...

2026-08-08T10:13:12Z

📖 Affine Algebraic Variety

An **affine algebraic variety** over a field $k$ is the set $V(I) = \\{a \\in k^n : f(a) = 0 \\text{ for all } f \\in I\\}$ where $I$ is an ideal in $k[x_1, \\ldots, x_n]$. The **coordinate ring** is $k[V] = k[x_1, \\ldots, x_n]/I(V)$.

From: gal-morandi
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📐 Fundamental Theorem of Symmetric Functions (Generalized) $S ...

2026-08-08T09:22:15Z

📐 Fundamental Theorem of Symmetric Functions (Generalized)

$S = F$ and $(E/F) = n!$. Any polynomial in $x_1, \\ldots, x_n$ can be uniquely expressed as a linear combination of $x_1^{\\nu_1} \\cdots x_n^{\\nu_n}$ (with $\\nu_i \\leq i-1$) with coefficients that are polynomials in $a_1, \\ldots, a_n$.

From: gal-artin
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📖 Fixed Point and Fixed Field An element $a \\in E$ such that ...

2026-08-08T05:26:06Z

📖 Fixed Point and Fixed Field

An element $a \\in E$ such that $\\sigma_1(a) = \\sigma_2(a) = \\cdots = \\sigma_n(a)$ is a fixed point. The set of all fixed points forms a subfield called the fixed field.

From: gal-artin
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📐 Galois Groups of Cubics An irreducible cubic $f(x) \\in ...

2026-08-08T02:35:03Z

📐 Galois Groups of Cubics

An irreducible cubic $f(x) \\in \\mathbb{Q}[x]$ has Galois group $S_3$ if $\\operatorname{disc}(f)$ is not a perfect square in $\\mathbb{Q}$, and $A_3 \\cong \\mathbb{Z}/3\\mathbb{Z}$ if it is.

From: gal-morandi
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📖 Algebraic Element An element $\\alpha \\in E$ is algebraic ...

2026-08-07T23:17:26Z

📖 Algebraic Element

An element $\\alpha \\in E$ is algebraic over $F$ if it is a root of some nonzero polynomial in $F[X]$. Otherwise, $\\alpha$ is transcendental over $F$.

From: gal-weintraub
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📐 Eigenvectors are Independent Let $T \\in \\mathcal{L}(V)$. ...

2026-08-07T21:22:12Z

📐 Eigenvectors are Independent

Let $T \\in \\mathcal{L}(V)$. Eigenvectors of $T$ corresponding to distinct eigenvalues are linearly independent.

From: linalg-axler
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📐 Resolvent is Solvable The resolvent equation of the cubic, ...

2026-08-07T19:39:00Z

📐 Resolvent is Solvable

The resolvent equation of the cubic, although of degree 6 in the variable $t$, is actually a quadratic equation in $t^3$. It can therefore be solved by first solving a quadratic (to find $t^3$) and then taking a cube root (to find $t$). Once $t$ is found, the original roots are determined.

From: gal-edwards
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📐 Galois Group of the Cyclotomic Equation The Galois group of ...

2026-08-07T18:29:55Z

📐 Galois Group of the Cyclotomic Equation

The Galois group of $x^p - 1 = 0$ over $\\mathbb{Q}$ (equivalently, the Galois group of $\\Phi_p(x) = 0$ over $\\mathbb{Q}$) is the cyclic group of order $p - 1$, isomorphic to $(\\mathbb{Z}/p\\mathbb{Z})^*$. Each automorphism sends $a$ to some power $a^j$ where $j \\not\\equiv 0 \\pmod{p}$.

From: gal-edwards
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💡 Proposition (Reduction by a Cyclic Extension) Consider the ...

2026-08-07T16:34:03Z

💡 Proposition (Reduction by a Cyclic Extension)

Consider the Galois group of $f(x) = 0$ over a field $K$. Let $p$ be a prime, let $K$ contain primitive $p$th roots of unity, and let $K\

Proof: Let $H(X)$ be the irreducible factor of $F(X)$ over $K\

From: gal-edwards
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📖 Perfect Field A field $F$ is perfect if every ...

2026-08-07T15:36:47Z

📖 Perfect Field

A field $F$ is **perfect** if every irreducible polynomial in $F[x]$ is separable. Every field of characteristic $0$ is perfect, and a field of characteristic $p$ is perfect if and only if $F = F^p$ (i.e., the Frobenius is surjective).

From: gal-morandi
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📖 Algebraic Closure An algebraic closure of $F$ is an ...

2026-08-07T12:27:58Z

📐 Division Algorithm For any two polynomials $f(x)$ and $g(x)$ ...

2026-08-07T11:31:37Z

📐 Division Algorithm

For any two polynomials $f(x)$ and $g(x)$ in $F$ with $g \\neq 0$: $f(x) = q(x) \\cdot g(x) + r(x)$ where $q(x)$ and $r(x)$ are unique and $\\deg(r) < \\deg(g)$.

Proof: Subtract suitable multiples of $g(x)$ from $f(x)$ to reduce the degree. Since the degree decreases at each step, the process terminates with $\\deg(r) < \\deg(g)$. Uniqueness: if $q_1 g + r_1 = q_2 g + r_2$, then $(q_1 - q_2)g = r_2 - r_1$, and the degree constraint forces $q_1 = q_2$, $r_1 = r_2$.

From: gal-artin
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📖 Irreducible Polynomial A polynomial $f \\in F[X]$ of degree ...

2026-08-07T10:33:34Z

📖 Irreducible Polynomial

A polynomial $f \\in F[X]$ of degree $\\geq 1$ is irreducible over $F$ if it cannot be written as a product $f = gh$ with $\\deg(g), \\deg(h) \\geq 1$.

From: gal-weintraub
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📐 Invertibility Characterization A linear map $T \\in ...

2026-08-07T06:53:17Z

📐 Invertibility Characterization

A linear map $T \\in \\mathcal{L}(V, W)$ is invertible if and only if $T$ is injective and surjective.

From: linalg-axler
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📖 Cyclic Extension A field extension $K\ From: gal-edwards ...

2026-08-07T00:31:06Z

📖 Cyclic Extension

A field extension $K\

From: gal-edwards
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📐 Division Algorithm For any two polynomials $f(x)$ and $g(x)$ ...

2026-08-06T13:19:48Z

🎮 Interactive: Quaternion Rotation Demo Visualize 3D rotations ...

2026-08-06T11:07:06Z

🎮 Interactive: Quaternion Rotation Demo

Visualize 3D rotations using quaternions. See why video game developers and aerospace engineers prefer quaternions over Euler angles.

From: Four Pillars of Geometry
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📐 Galois A polynomial equation with rational coefficients is ...

2026-08-06T08:06:27Z

📐 Galois

A polynomial equation with rational coefficients is solvable by radicals if and only if its Galois group is a solvable group. In particular, for equations of degree 5 or higher, the Galois group may fail to be solvable, which is why no general radical formula exists.

Proof: The proof of this theorem is the goal of the entire book. It requires developing the theory of Galois groups, the connection between field extensions and group theory, and the concept of solvable groups. The full proof appears in Part 6 of Edwards\

From: gal-edwards
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🎯 Corollary (Isomorphism of Splitting Fields) Any two ...

2026-08-06T04:48:53Z

🎯 Corollary (Isomorphism of Splitting Fields)

Any two splitting fields for $p(x)$ over $F$ are isomorphic.

Proof: Apply Theorem 10 with $F = F\

From: gal-artin
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📐 Cauchy-Schwarz Inequality If $u, v \\in V$ (inner product ...

2026-08-06T01:05:53Z

📐 Cauchy-Schwarz Inequality

If $u, v \\in V$ (inner product space), then $|\\langle u, v \\rangle| \\leq \\|u\\| \\|v\\|$. Equality holds iff one is a scalar multiple of the other.

Proof: If $v = 0$, both sides are 0. Otherwise, let $c = \\langle u, v \\rangle / \\|v\\|^2$. Then $0 \\leq \\|u - cv\\|^2 = \\|u\\|^2 - |\\langle u,v\\rangle|^2/\\|v\\|^2$, giving the result.

From: linalg-axler
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📐 Newton Any symmetric polynomial in the roots of an equation ...

2026-08-05T21:56:05Z

📐 Newton

Any symmetric polynomial in the roots of an equation can be expressed in terms of the coefficients of that equation. That is, symmetric functions of the roots are computable without finding the roots themselves.

Proof: The formulas follow immediately from the identity $x^3 + bx^2 + cx + d = (x - r)(x - s)(x - t)$. When the right side is multiplied out and coefficients of like powers of $x$ are equated, one obtains: $r + s + t = -b$, $rs + rt + st = c$, $rst = -d$. Any other symmetric function can be built from ...

From: gal-edwards
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📖 Normal Subgroup A subgroup $H$ of a group $G$ is called ...

2026-08-05T20:14:30Z

📖 Normal Subgroup

A subgroup $H$ of a group $G$ is called normal if for every $S$ in $H$ and every $T$ in $G$, the conjugate $T^{-1}ST$ is also in $H$. Equivalently, $H$ is normal if the various coset presentations of the subgroup differ from one another by a single substitution.

From: gal-edwards
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📖 Solution to Noether A system $\\{x_\\sigma\\}$ indexed by ...

2026-08-05T18:11:25Z

📖 Solution to Noether

A system $\\{x_\\sigma\\}$ indexed by elements of a group $G$ of automorphisms of $E$ is a solution to Noether\

From: gal-artin
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📐 Insolvability of the General Quintic The general polynomial ...

2026-08-05T04:48:34Z

📐 Insolvability of the General Quintic

The general polynomial of degree $n \\geq 5$ is not solvable by radicals. The symmetric group $S_n$ for $n \\geq 5$ is not solvable, since $A_n$ is simple for $n \\geq 5$.

From: gal-morandi
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🎯 Corollary (Isomorphism of Splitting Fields) Any two ...

2026-08-04T23:45:13Z

📐 Tower Law If $K \\subseteq L \\subseteq M$ are fields, then ...

2026-08-04T22:44:38Z

📐 Tower Law

If $K \\subseteq L \\subseteq M$ are fields, then $[M:K] = [M:L] \\cdot [L:K]$. In particular, $[M:K]$ is finite if and only if both $[M:L]$ and $[L:K]$ are finite.

Proof: Let $\\{e_1, \\ldots, e_m\\}$ be a basis for $M/L$ and $\\{f_1, \\ldots, f_n\\}$ a basis for $L/K$. Then $\\{e_i f_j\\}$ is a basis for $M/K$, giving $[M:K] = mn = [M:L][L:K]$.

From: gal-jacobson
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📐 Theorem 19 (Artin) A polynomial $f$ has repeated roots if ...

2026-08-04T20:55:43Z

📐 Theorem 19 (Artin)

A polynomial $f$ has repeated roots if and only if $f$ and $f\

From: gal-artin
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📐 Galois A polynomial equation with rational coefficients is ...

2026-08-04T19:14:27Z

📐 Newton\ The power sums $s_k = r_1^k + r_2^k + \\cdots + ...

2026-08-04T17:21:08Z

📐 Newton\

The power sums $s_k = r_1^k + r_2^k + \\cdots + r_n^k$ satisfy the recurrence relation: $s_k - s_{k-1}\\sigma_1 + s_{k-2}\\sigma_2 - \\cdots + (-1)^{k-1}s_1\\sigma_{k-1} + (-1)^k k\\sigma_k = 0$ for $k = 1, 2, 3, \\ldots$, where $\\sigma_j = 0$ for $j > n$.

Proof: This recurrence follows from the identity $r_i^n - \\sigma_1 r_i^{n-1} + \\sigma_2 r_i^{n-2} - \\cdots \\pm \\sigma_n = 0$, which holds for each root $r_i$. Summing over $i$ and using the definition of the power sums gives $s_n - \\sigma_1 s_{n-1} + \\sigma_2 s_{n-2} - \\cdots \\pm n\\sigma_n = 0...

From: gal-edwards
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📖 Krull Topology The Krull topology on ...

2026-08-04T12:29:35Z

📖 Krull Topology

The **Krull topology** on $\\operatorname{Gal}(K/F)$ for an (infinite) Galois extension has as basic open sets the cosets $\\sigma \\cdot \\operatorname{Gal}(K/L)$ where $L/F$ is a finite Galois sub-extension. With this topology, $\\operatorname{Gal}(K/F)$ is a profinite group.

From: gal-morandi
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🎮 Interactive: RSA Encryption Demo See how RSA public-key ...

2026-08-04T10:18:51Z

🎮 Interactive: RSA Encryption Demo

See how RSA public-key encryption works. The security relies on the difficulty of factoring large numbers!

From: Cryptography Math
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🎯 Corollary (Isomorphism of Splitting Fields) Any two ...

2026-08-04T03:54:34Z

📐 Primitive Element Theorem If $E/F$ is a finite separable ...

2026-08-03T23:48:19Z

📐 Primitive Element Theorem

If $E/F$ is a finite separable extension, then $E = F(\\alpha)$ for some $\\alpha \\in E$.

Proof: If $F$ is infinite, let $E = F(\\alpha, \\beta)$. Choose $c \\in F$ so that $\\gamma = \\alpha + c\\beta$ is a primitive element. This is possible since only finitely many values of $c$ fail. The key step is showing $\\beta \\in F(\\gamma)$ by analyzing gcd arguments with minimal polynomials.

From: gal-weintraub
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📖 Transcendence Base and Degree A subset $S$ of $K$ is ...

2026-08-03T20:47:12Z

📖 Transcendence Base and Degree

A subset $S$ of $K$ is **algebraically independent** over $F$ if no element of $S$ is algebraic over $F(S \\setminus \\{s\\})$ for any $s \\in S$. A **transcendence base** is a maximal algebraically independent set. The **transcendence degree** $\\operatorname{tr.deg}(K/F)$ is the cardinality of any transcendence base.

From: gal-morandi
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📐 Theorem 7 (Kronecker) If $f(x)$ is a polynomial in a field ...

2026-08-03T19:10:40Z

📐 Theorem 7 (Kronecker)

If $f(x)$ is a polynomial in a field $F$, there exists an extension field $E$ of $F$ in which $f(x)$ has a root.

Proof: Factor $f(x)$ into irreducible factors. For an irreducible factor of degree $n$, construct $E_1 = F[\\xi]/(f(\\xi))$, the set of formal polynomials in a symbol $\\xi$ of degree $< n$, with multiplication defined modulo $f(\\xi)$. The irreducibility of $f$ guarantees that $E_1$ is a field in which...

From: gal-artin
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🔗 Lemma 1 (LCM of Orders) In an abelian group, if $A$ and $B$ ...

2026-08-03T17:28:34Z

🔗 Lemma 1 (LCM of Orders)

In an abelian group, if $A$ and $B$ have orders $a$ and $b$ with $\\mathrm{lcm}(a,b) = c$, then there exists an element of order $c$.

From: gal-artin
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📖 Solving by Radicals An equation is said to be solvable by ...

2026-08-03T15:32:41Z

📖 Solving by Radicals

An equation is said to be solvable by radicals if its roots can be expressed in terms of its coefficients using only the operations of addition, subtraction, multiplication, division, and the extraction of $n$th roots for various $n$.

From: gal-edwards
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📐 Dedekind Distinct characters $\\chi_1, \\ldots, \\chi_n: G ...

2026-08-03T08:51:10Z

📐 Dedekind

Distinct characters $\\chi_1, \\ldots, \\chi_n: G \\to K^\\times$ of a group $G$ are linearly independent over $K$. That is, if $\\sum a_i \\chi_i = 0$ for $a_i \\in K$, then all $a_i = 0$.

From: gal-jacobson
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📐 Characterization of Cyclic Extensions Let $F$ contain a ...

2026-08-03T05:22:27Z

📐 Characterization of Cyclic Extensions

Let $F$ contain a primitive $n$th root of unity. Then $K/F$ is cyclic of degree $n$ if and only if $K = F(\\alpha)$ where $\\alpha^n \\in F$.

From: gal-morandi
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📐 Frobenius Automorphism The Galois group ...

2026-08-02T23:42:23Z

📐 Frobenius Automorphism

The Galois group $\\operatorname{Gal}(\\mathbb{F}_{p^n}/\\mathbb{F}_p)$ is cyclic of order $n$, generated by the Frobenius automorphism $\\phi: a \\mapsto a^p$.

From: gal-morandi
Learn more: https://mathacademy-cyan.vercel.app/#/section/5

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📐 Fundamental Theorem of Symmetric Functions (Generalized) $S ...

2026-08-02T21:37:28Z

📐 Simple Algebraic Extensions Let $K$ be a given field and let ...

2026-08-02T17:44:45Z

📐 Simple Algebraic Extensions

Let $K$ be a given field and let $G(X)$ be an irreducible polynomial with coefficients in $K$. Then one can construct a field $K(t)$ such that: (1) $K(t)$ contains $K$, (2) $K(t)$ contains an element $t$ with $G(t) = 0$, and (3) every element of $K(t)$ can be expressed as a polynomial $b_0 + b_1 t + \\cdots + b_\\nu t^\\nu$ where $\\nu < \\deg G$. Moreover, any two such fields are naturally iso...

Proof: Let $R$ be the set of all polynomials in $X$ with coefficients in $K$. Two elements are congruent mod $G$ if their difference is divisible by $G(X)$. The quotient $L$ is a ring. The mapping $k \\mapsto$ [class of constant $k$] embeds $K$ into $L$. The class of $X$ is a root of $G$ in $L$. The Euc...

From: gal-edwards
Learn more: https://mathacademy-cyan.vercel.app/#/section/11

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📐 Abel\u2013Ruffini Theorem The general polynomial equation of ...

2026-08-02T16:39:31Z

📐 Abel\u2013Ruffini Theorem

The general polynomial equation of degree $n \\geq 5$ is not solvable by radicals. This follows from the fact that $S_n$ is not solvable for $n \\geq 5$.

From: gal-jacobson
Learn more: https://mathacademy-cyan.vercel.app/#/section/11

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📐 Sufficient Condition for Diagonalizability If $T \\in ...

2026-08-02T15:45:01Z

📐 Sufficient Condition for Diagonalizability

If $T \\in \\mathcal{L}(V)$ has $\\dim V$ distinct eigenvalues, then $T$ is diagonalizable.

From: linalg-axler
Learn more: https://mathacademy-cyan.vercel.app/#/section/15

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📖 Derivation A derivation from a ring $R$ to an $R$-module ...

2026-08-02T12:43:52Z

📖 Derivation

A **derivation** from a ring $R$ to an $R$-module $M$ is a map $D: R \\to M$ satisfying $D(ab) = aD(b) + bD(a)$ (the Leibniz rule). The module of **K\\u00E4hler differentials** $\\Omega_{K/F}$ is the universal target for $F$-derivations from $K$.

From: gal-morandi
Learn more: https://mathacademy-cyan.vercel.app/#/section/22

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📐 Solution by Radicals (Full Version) Let $f(x) = 0$ be an ...

2026-08-01T23:41:28Z

📐 Solution by Radicals (Full Version)

Let $f(x) = 0$ be an equation with coefficients in a field $K$ (obtained from $\\mathbb{Q}$ by a finite number of adjunctions). A solution by radicals is a sequence of field extensions $K \\subset K_1 \\subset \\cdots \\subset K_\\mu$ where each $K_i$ is obtained by adjoining a $p_i$th root of an element of $K_{i-1}$ (with $p_i$th roots of unity present in $K_{i-1}$). Such a solution exists if ...

From: gal-edwards
Learn more: https://mathacademy-cyan.vercel.app/#/section/21

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📐 Gauss Let $d$ and $D$ be divisors of $p - 1$, with $d | D$ ...

2026-08-01T22:39:08Z

📐 Gauss

Let $d$ and $D$ be divisors of $p - 1$, with $d | D$ and $D/d = q$. Then $K_D$ is a simple algebraic extension of $K_d$ of degree $q$. Any given element of $K_D$ can be expressed rationally in terms of elements of $K_d$, a $q$th root of an element of $K_d$, and a primitive $q$th root of unity.

Proof: The Lagrange resolvent $t = \\gamma + \\beta \\cdot S^d\\gamma + \\beta^2 \\cdot S^{2d}\\gamma + \\cdots + \\beta^{q-1} \\cdot S^{(q-1)d}\\gamma$ where $\\beta$ is a primitive $q$th root of unity satisfies $t^q \\in K_d$, so the extension from $K_d$ to $K_D$ requires only a $q$th root.

From: gal-edwards
Learn more: https://mathacademy-cyan.vercel.app/#/section/8

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📖 Normal Extension A finite extension $E/F$ is normal if the ...

2026-08-01T21:37:18Z

📖 Normal Extension

A finite extension $E/F$ is normal if the group $G$ of automorphisms of $E$ fixing $F$ has $F$ as its fixed field. Equivalently, $F$ is exactly the set of elements fixed by all automorphisms leaving $F$ fixed.

From: gal-artin
Learn more: https://mathacademy-cyan.vercel.app/#/section/13

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📐 Existence of Splitting Fields Every polynomial $f \\in F[X]$ ...

2026-08-01T18:42:03Z

📐 Existence of Splitting Fields

Every polynomial $f \\in F[X]$ of degree $\\geq 1$ has a splitting field, and any two splitting fields of $f$ over $F$ are isomorphic via an isomorphism fixing $F$.

From: gal-weintraub
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📐 Classical Impossibility Results It is impossible to (1) ...

2026-08-01T17:42:37Z

📐 Classical Impossibility Results

It is impossible to (1) double the cube ($\\sqrt[3]{2}$ is not constructible since $[\\mathbb{Q}(\\sqrt[3]{2}):\\mathbb{Q}] = 3$), (2) trisect a general angle, or (3) square the circle ($\\pi$ is transcendental).

From: gal-morandi
Learn more: https://mathacademy-cyan.vercel.app/#/section/14

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🎮 Interactive: Angle Preservation in Hyperbolic Geometry See ...

2026-08-01T07:51:37Z

🎮 Interactive: Angle Preservation in Hyperbolic Geometry

See how the hyperbolic plane preserves angles but distorts distances. Maps that preserve angles are called conformal.

From: Four Pillars of Geometry
Try it: https://four-pillars-deploy.vercel.app/#/section/60

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📖 Algebraic Element An element $\\alpha \\in E$ is algebraic ...

2026-08-01T05:00:09Z

📐 Existence of Galois Resolvents For any polynomial equation ...

2026-08-01T01:15:38Z

📐 Existence of Galois Resolvents

For any polynomial equation of degree $n$ with distinct roots, there exist integers $A, B, C, \\ldots$ such that $t = Aa + Bb + Cc + \\cdots$ has $n!$ distinct values under all $n!$ permutations of the roots.

From: gal-edwards
Learn more: https://mathacademy-cyan.vercel.app/#/section/10

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📐 Unique Factorization for Polynomials over $\\mathbb{Z}$ A ...

2026-07-10T04:39:54Z

📐 Unique Factorization for Polynomials over $\\mathbb{Z}$

A representation of a polynomial with integer coefficients as a product of irreducibles is unique up to the order of the factors and their signs. That is, if $F_1 F_2 \\cdots F_\\mu = G_1 G_2 \\cdots G_\\nu$ where all factors are irreducible, then $\\mu = \\nu$ and the $G_j$\

From: gal-edwards
Learn more: https://mathacademy-cyan.vercel.app/#/section/18

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📖 Elementary Symmetric Polynomials The $k$th elementary ...

2026-07-09T23:50:29Z

📖 Elementary Symmetric Polynomials

The $k$th elementary symmetric polynomial $\\sigma_k$ in variables $r_1, r_2, \\ldots, r_n$ is the sum of all products of $k$ distinct variables chosen from the $r_i$. The relationships $\\sigma_k = (-1)^k b_k$ connect the elementary symmetric polynomials to the coefficients of the equation.

From: gal-edwards
Learn more: https://mathacademy-cyan.vercel.app/#/section/2

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📐 Theorem 10 (Uniqueness of Splitting Fields) An isomorphism ...

2026-07-09T22:08:21Z

📐 Theorem 10 (Uniqueness of Splitting Fields)

An isomorphism $\\sigma: F \\to F\

Proof: Induction on the number of roots outside $F$. Use Theorem 8 to extend $\\sigma$ to $F(\\alpha) \\to F\

From: gal-artin
Learn more: https://mathacademy-cyan.vercel.app/#/section/9

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📖 The Babylonian Normal Form The Babylonians commonly solved ...

2026-07-09T20:28:53Z

📖 The Babylonian Normal Form

The Babylonians commonly solved quadratic equations by reducing them to a normal form: given two numbers $p$ and $s$, find two numbers $x$ and $y$ such that $xy = p$ and $x + y = s$. This is equivalent to solving $x^2 - sx + p = 0$.

From: gal-edwards
Learn more: https://mathacademy-cyan.vercel.app/#/section/1

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📐 Theorem 4 The right column rank, left column rank, right row ...

2026-07-09T18:27:12Z

📐 Theorem 4

The right column rank, left column rank, right row rank, and left row rank of a matrix are all equal.

Proof: Show $c \\leq r$ by truncating to the first $r$ independent rows (the column rank does not change). Applying the same argument to the transpose gives $r \\leq c$, hence $r = c$. The same reasoning equates all four rank notions.

From: gal-artin
Learn more: https://mathacademy-cyan.vercel.app/#/section/3

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🎮 Interactive: Non-Euclidean Lines Demo Explore straight lines ...

2026-07-09T16:16:01Z

🎮 Interactive: Non-Euclidean Lines Demo

Explore straight lines in hyperbolic geometry. In the Poincare disk, geodesics appear as circular arcs perpendicular to the boundary.

From: Four Pillars of Geometry
Try it: https://four-pillars-deploy.vercel.app/#/section/61

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📐 Trace is Basis-Independent $\\operatorname{trace} T = ...

2026-07-09T13:36:24Z

📐 Trace is Basis-Independent

$\\operatorname{trace} T = \\operatorname{trace} \\mathcal{M}(T)$ for any basis, where the trace of a matrix is the sum of its diagonal entries.

From: linalg-axler
Learn more: https://mathacademy-cyan.vercel.app/#/section/29

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📖 Symmetric Polynomial in Roots Let $r, s, t$ be the three ...

2026-07-09T10:14:29Z

📖 Symmetric Polynomial in Roots

Let $r, s, t$ be the three roots of a cubic equation $x^3 + bx^2 + cx + d = 0$. The elementary symmetric polynomials are: $r + s + t = -b$, $rs + rt + st = c$, and $rst = -d$. Any symmetric polynomial in the roots can be expressed in terms of these.

From: gal-edwards
Learn more: https://mathacademy-cyan.vercel.app/#/section/2

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📐 Theorem 1 A system of $m$ homogeneous linear equations in ...

2026-07-09T05:53:17Z

📐 Theorem 1

A system of $m$ homogeneous linear equations in $n$ unknowns over a field $F$, with $n > m$, always has a non-trivial solution.

Proof: By induction on $m$. For $m = 0$, all unknowns are free. For the inductive step, use elimination: assuming $a_{11} \\neq 0$, form $m-1$ equations in $n-1 > m-1$ unknowns by subtracting multiples of the first equation. The inductive hypothesis gives a non-trivial solution, which extends to the ful...

From: gal-artin
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📖 Adjoint If $T \\in \\mathcal{L}(V, W)$, the adjoint $T^* ...

2026-07-09T01:20:07Z

📖 Adjoint

If $T \\in \\mathcal{L}(V, W)$, the adjoint $T^* \\in \\mathcal{L}(W, V)$ is the unique operator such that $\\langle Tv, w \\rangle = \\langle v, T^*w \\rangle$ for all $v \\in V$, $w \\in W$.

From: linalg-axler
Learn more: https://mathacademy-cyan.vercel.app/#/section/19

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📖 Extension Field If $E$ is a field and $F$ is a subset of $E$ ...

2026-07-08T23:45:25Z

📖 Extension Field

If $E$ is a field and $F$ is a subset of $E$ which itself forms a field under the operations of $E$, then $F$ is a subfield of $E$ and $E$ is an extension of $F$.

From: gal-artin
Learn more: https://mathacademy-cyan.vercel.app/#/section/6

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📐 Galois Let $f(x) = 0$ be an equation with distinct roots ...

2026-07-08T21:48:19Z

📐 Galois

Let $f(x) = 0$ be an equation with distinct roots whose Galois group over $K$ is $G$. Then $f(x) = 0$ can be solved by radicals if and only if $G$ is solvable -- that is, has a composition series $G \\supset G_1 \\supset G_2 \\supset \\cdots \\supset G_\\nu = \\{e\\}$ in which each $G_i$ is a normal subgroup of prime index in its predecessor.

Proof: Necessity: If solvable by radicals, the tower of field extensions reduces the Galois group at each step to a normal subgroup of prime index (by the proposition of \u00a744). Taking only steps where the group decreases gives the composition series. Sufficiency: If $G$ is solvable, the proposition ...

From: gal-edwards
Learn more: https://mathacademy-cyan.vercel.app/#/section/16

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📖 Separable Polynomial and Separable Extension A polynomial is ...

2026-07-08T20:09:20Z

📖 Separable Polynomial and Separable Extension

A polynomial is separable if its irreducible factors have no repeated roots. An element is separable if it is a root of a separable polynomial. The extension $E/F$ is separable if every element of $E$ is separable over $F$.

From: gal-artin
Learn more: https://mathacademy-cyan.vercel.app/#/section/13

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📐 Theorem 4 The right column rank, left column rank, right row ...

2026-07-08T18:06:33Z

📐 Trace is Basis-Independent $\\operatorname{trace} T = ...

2026-07-08T15:53:44Z

📖 Lagrange Resolvent for the Cyclotomic Equation For the ...

2026-07-08T13:15:57Z

📖 Lagrange Resolvent for the Cyclotomic Equation

For the equation $x^{p} - 1 = 0$ (with $p$ prime), let $\\beta$ be a primitive $(p-1)$st root of unity. The Lagrange resolvent is $t = \\alpha^j + \\beta \\alpha^k + \\beta^2 \\alpha^m + \\cdots$ where the roots are listed in the order determined by a primitive root modulo $p$. The quantity $t^{p-1}$ is a known quantity, expressible in terms of $\\beta$ alone.

From: gal-edwards
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📖 Primitive Root of Unity An $n$th root of unity is a number ...

2026-07-08T10:49:18Z

📖 Primitive Root of Unity

An $n$th root of unity is a number $\\alpha$ satisfying $\\alpha^n = 1$. It is primitive if no smaller positive power of $\\alpha$ equals 1. For $n = 3$, the primitive cube roots of unity are $\\alpha = (-1 \\pm \\sqrt{-3})/2$.

From: gal-edwards
Learn more: https://mathacademy-cyan.vercel.app/#/section/4

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📐 Norm and Trace via Galois Group If $K/F$ is Galois with ...

2026-07-08T07:58:40Z

📐 Norm and Trace via Galois Group

If $K/F$ is Galois with group $G$, then $N_{K/F}(\\alpha) = \\prod_{\\sigma \\in G} \\sigma(\\alpha)$ and $T_{K/F}(\\alpha) = \\sum_{\\sigma \\in G} \\sigma(\\alpha)$.

From: gal-morandi
Learn more: https://mathacademy-cyan.vercel.app/#/section/7

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📐 Kummer Theory Let $K$ be a field containing a primitive ...

2026-07-08T01:11:20Z

📐 Kummer Theory

Let $K$ be a field containing a primitive $n$-th root of unity $\\zeta_n$ (with $\\mathrm{char}(K) \\nmid n$). There is a bijection between cyclic extensions $L/K$ of degree dividing $n$ and subgroups of $K^\\times/(K^\\times)^n$, given by $L = K(a^{1/n})$ for $a \\in K^\\times$.

From: gal-jacobson
Learn more: https://mathacademy-cyan.vercel.app/#/section/18

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🔗 Lemma 2 (Maximal Order Divides All) If $C$ has maximal order ...

2026-07-07T23:36:45Z

🔗 Lemma 2 (Maximal Order Divides All)

If $C$ has maximal order $c$ in an abelian group, then $c$ is divisible by the order of every element, so $x^c = 1$ for all elements.

From: gal-artin
Learn more: https://mathacademy-cyan.vercel.app/#/section/14

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📐 Trace is Basis-Independent $\\operatorname{trace} T = ...

2026-07-07T22:00:59Z

📖 Trace If $T \\in \\mathcal{L}(V)$ and $\\lambda_1, \\ldots, ...

2026-07-07T20:31:38Z

📖 Trace

If $T \\in \\mathcal{L}(V)$ and $\\lambda_1, \\ldots, \\lambda_n$ are the eigenvalues of $T$ (counted with multiplicity), then $\\operatorname{trace} T = \\lambda_1 + \\cdots + \\lambda_n$.

From: linalg-axler
Learn more: https://mathacademy-cyan.vercel.app/#/section/29

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📖 Dimension The dimension of a finite-dimensional vector space ...

2026-07-07T18:36:08Z

📖 Dimension

The dimension of a finite-dimensional vector space is the length of any basis of the vector space, denoted $\\dim V$.

From: linalg-axler
Learn more: https://mathacademy-cyan.vercel.app/#/section/5

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📖 Lagrange Resolvent Given an equation of degree $n$ with ...

2026-07-07T13:05:04Z

📖 Lagrange Resolvent

Given an equation of degree $n$ with roots $x_1, x_2, \\ldots, x_n$, a Lagrange resolvent is a quantity of the form $t = x_1 + \\alpha x_2 + \\alpha^2 x_3 + \\cdots + \\alpha^{n-1} x_n$ where $\\alpha$ is an $n$th root of unity (not necessarily primitive).

From: gal-edwards
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📖 Normal Extension An algebraic extension $K/F$ is normal ...

2026-07-07T10:14:57Z

📖 Normal Extension

An algebraic extension $K/F$ is **normal** if every irreducible polynomial in $F[x]$ that has a root in $K$ splits completely in $K[x]$. Equivalently, $K$ is a splitting field of a family of polynomials over $F$.

From: gal-morandi
Learn more: https://mathacademy-cyan.vercel.app/#/section/2

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📐 Simple Algebraic Extensions Let $K$ be a given field and let ...

2026-07-07T05:51:16Z

📖 Character of a Group A homomorphism $\\sigma$ from a ...

2026-07-07T01:24:13Z

📖 Character of a Group

A homomorphism $\\sigma$ from a multiplicative group $G$ into a field $F$ (with $\\sigma(\\alpha) \\neq 0$ for all $\\alpha$) is called a character of $G$ in $F$.

From: gal-artin
Learn more: https://mathacademy-cyan.vercel.app/#/section/11

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📖 Splitting Field A splitting field of $f \\in F[X]$ is an ...

2026-07-06T22:47:30Z

📖 Splitting Field

A splitting field of $f \\in F[X]$ is an extension $E/F$ such that $f$ splits completely in $E[X]$ and $E$ is generated over $F$ by the roots of $f$.

From: gal-weintraub
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📐 Division Algorithm for Polynomials If $f, g \\in F[X]$ with ...

2026-07-06T21:14:36Z

📐 Division Algorithm for Polynomials

If $f, g \\in F[X]$ with $g \\neq 0$, there exist unique $q, r \\in F[X]$ with $f = qg + r$ and $\\deg(r) < \\deg(g)$.

From: gal-weintraub
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📐 Gram-Schmidt Procedure If $v_1, \\ldots, v_m$ is a linearly ...

2026-07-06T19:23:15Z

📖 Norm and Trace For a finite extension $K/F$ and $\\alpha ...

2026-07-06T17:30:39Z

📖 Norm and Trace

For a finite extension $K/F$ and $\\alpha \\in K$, the **norm** $N_{K/F}(\\alpha) = \\det(L_\\alpha)$ and the **trace** $T_{K/F}(\\alpha) = \\operatorname{tr}(L_\\alpha)$, where $L_\\alpha: K \\to K$ is left multiplication by $\\alpha$.

From: gal-morandi
Learn more: https://mathacademy-cyan.vercel.app/#/section/7

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📖 Simple Group A group $G$ is called simple if it has no ...

2026-07-06T11:16:45Z

📖 Simple Group

A group $G$ is called simple if it has no normal subgroups other than $\\{e\\}$ and $G$ itself. A simple group of order greater than 1 is an obstruction to solvability: if it appears as a quotient in any composition series, and it is not of prime order, then the group is not solvable. The alternating group $A_5$ (with 60 elements) is the smallest non-abelian simple group.

From: gal-edwards
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🎮 Interactive: Riemann Sum Visualizer Approximate area under a ...

2026-07-06T06:10:22Z

🎮 Interactive: Riemann Sum Visualizer

Approximate area under a curve with rectangles. Watch how the limit of Riemann sums defines the integral.

From: Calculus I
Try it: https://calc1-course.vercel.app/#/section/10

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🎮 Interactive: Eigenvalue Explorer Find vectors that only get ...

2026-07-06T01:26:41Z

🎮 Interactive: Eigenvalue Explorer

Find vectors that only get scaled by a matrix. Eigenvalues reveal the fundamental structure of linear transformations.

From: Linear Algebra
Try it: https://linalg-pink.vercel.app/#/section/10

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📐 Sample Theorem If $A \\subseteq B$ and $B \\subseteq A$, ...

2026-07-05T23:44:33Z

📐 Sample Theorem

If $A \\subseteq B$ and $B \\subseteq A$, then $A = B$

Proof: Let $x \\in A$. Since $A \\subseteq B$, we have $x \\in B$ by definition of subset.
Therefore, every element of $A$ is in $B$.

Now, let $y \\in B$. Since $B \\subseteq A$, we have $y \\in A$ by definition.
Therefore, every element of $B$ is in $A$.

Since $A \\subseteq B$ and $B \\subseteq A...

From: gal-howie
Learn more: https://mathacademy-cyan.vercel.app/#/section/2

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Nostr notes by MagicInternetMath

📖 Algebraic Closure An **algebraic closure** of $F$ is an ...

📖 Mining Pool A mining pool aggregates many miners\ From: ...

📖 PPS / FPPS Pay-per-share: the pool pays a fixed price for ...

📐 Bonferroni Bound Testing $m$ cells at once, requiring ...

📖 Roots of Unity The roots of $x^n - 1$ form a cyclic group ...

📖 Resolvent Equation The auxiliary equation of lower degree ...

📐 Characteristic is Prime The characteristic of a field is ...

📖 Linear Substitution Modulo q A permutation $\\sigma$ of ...

📖 Galois Group The Galois group of the equation $f(x) = 0$ ...

📐 Gram-Schmidt Procedure If $v_1, \\ldots, v_m$ is a linearly ...

🎮 Interactive: Determinant Visualizer Understand the ...

📐 Fundamental Theorem of Galois Theory Let $E/K$ be a finite ...

📖 Affine Algebraic Variety An **affine algebraic variety** ...

📐 Fundamental Theorem of Symmetric Functions (Generalized) $S ...

📖 Fixed Point and Fixed Field An element $a \\in E$ such that ...

📐 Galois Groups of Cubics An irreducible cubic $f(x) \\in ...

📖 Algebraic Element An element $\\alpha \\in E$ is algebraic ...

📐 Eigenvectors are Independent Let $T \\in \\mathcal{L}(V)$. ...

📐 Resolvent is Solvable The resolvent equation of the cubic, ...

📐 Galois Group of the Cyclotomic Equation The Galois group of ...

💡 Proposition (Reduction by a Cyclic Extension) Consider the ...

📖 Perfect Field A field $F$ is **perfect** if every ...

📖 Algebraic Closure An **algebraic closure** of $F$ is an ...

📐 Division Algorithm For any two polynomials $f(x)$ and $g(x)$ ...

📖 Irreducible Polynomial A polynomial $f \\in F[X]$ of degree ...

📐 Invertibility Characterization A linear map $T \\in ...

📖 Cyclic Extension A field extension $K\ From: gal-edwards ...

📐 Division Algorithm For any two polynomials $f(x)$ and $g(x)$ ...

🎮 Interactive: Quaternion Rotation Demo Visualize 3D rotations ...

📐 Galois A polynomial equation with rational coefficients is ...

🎯 Corollary (Isomorphism of Splitting Fields) Any two ...

📐 Cauchy-Schwarz Inequality If $u, v \\in V$ (inner product ...

📐 Newton Any symmetric polynomial in the roots of an equation ...

📖 Normal Subgroup A subgroup $H$ of a group $G$ is called ...

📖 Solution to Noether A system $\\{x_\\sigma\\}$ indexed by ...

📐 Insolvability of the General Quintic The general polynomial ...

🎯 Corollary (Isomorphism of Splitting Fields) Any two ...

📐 Tower Law If $K \\subseteq L \\subseteq M$ are fields, then ...

📐 Theorem 19 (Artin) A polynomial $f$ has repeated roots if ...

📐 Galois A polynomial equation with rational coefficients is ...

📐 Newton\ The power sums $s_k = r_1^k + r_2^k + \\cdots + ...

📖 Krull Topology The **Krull topology** on ...

🎮 Interactive: RSA Encryption Demo See how RSA public-key ...

🎯 Corollary (Isomorphism of Splitting Fields) Any two ...

📐 Primitive Element Theorem If $E/F$ is a finite separable ...

📖 Transcendence Base and Degree A subset $S$ of $K$ is ...

📐 Theorem 7 (Kronecker) If $f(x)$ is a polynomial in a field ...

🔗 Lemma 1 (LCM of Orders) In an abelian group, if $A$ and $B$ ...

📖 Solving by Radicals An equation is said to be solvable by ...

📐 Dedekind Distinct characters $\\chi_1, \\ldots, \\chi_n: G ...

📐 Characterization of Cyclic Extensions Let $F$ contain a ...

📐 Frobenius Automorphism The Galois group ...

📐 Fundamental Theorem of Symmetric Functions (Generalized) $S ...

📐 Simple Algebraic Extensions Let $K$ be a given field and let ...

📐 Abel\u2013Ruffini Theorem The general polynomial equation of ...

📐 Sufficient Condition for Diagonalizability If $T \\in ...

📖 Derivation A **derivation** from a ring $R$ to an $R$-module ...

📐 Solution by Radicals (Full Version) Let $f(x) = 0$ be an ...

📐 Gauss Let $d$ and $D$ be divisors of $p - 1$, with $d | D$ ...

📖 Normal Extension A finite extension $E/F$ is normal if the ...

📐 Existence of Splitting Fields Every polynomial $f \\in F[X]$ ...

📐 Classical Impossibility Results It is impossible to (1) ...

🎮 Interactive: Angle Preservation in Hyperbolic Geometry See ...

📖 Algebraic Element An element $\\alpha \\in E$ is algebraic ...

📐 Existence of Galois Resolvents For any polynomial equation ...

📐 Unique Factorization for Polynomials over $\\mathbb{Z}$ A ...

📖 Elementary Symmetric Polynomials The $k$th elementary ...

📐 Theorem 10 (Uniqueness of Splitting Fields) An isomorphism ...

📖 The Babylonian Normal Form The Babylonians commonly solved ...

📐 Theorem 4 The right column rank, left column rank, right row ...

🎮 Interactive: Non-Euclidean Lines Demo Explore straight lines ...

📐 Trace is Basis-Independent $\\operatorname{trace} T = ...

📖 Symmetric Polynomial in Roots Let $r, s, t$ be the three ...

📐 Theorem 1 A system of $m$ homogeneous linear equations in ...

📖 Adjoint If $T \\in \\mathcal{L}(V, W)$, the adjoint $T^* ...

📖 Extension Field If $E$ is a field and $F$ is a subset of $E$ ...

📐 Galois Let $f(x) = 0$ be an equation with distinct roots ...

📖 Separable Polynomial and Separable Extension A polynomial is ...

📐 Theorem 4 The right column rank, left column rank, right row ...

📖 Algebraic Closure An algebraic closure of $F$ is an ...

📖 Affine Algebraic Variety An affine algebraic variety ...

📖 Perfect Field A field $F$ is perfect if every ...

📖 Algebraic Closure An algebraic closure of $F$ is an ...

📖 Krull Topology The Krull topology on ...

📖 Derivation A derivation from a ring $R$ to an $R$-module ...

📖 Normal Extension An algebraic extension $K/F$ is normal ...