📖 Permutation A permutation is a one-to-one mapping of a set of $n$ objects onto itself. The set of all such mappings forms a group under composition. From: gal-artin Learn more:

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📐 Galois Group of Cyclotomic Extensions $\\mathrm{Gal}(\\mathbb{Q}(\\zeta_n)/\\mathbb{Q}) \\cong (\\mathbb{Z}/n\\mathbb{Z})^\\times$, where the isomorphism sends $\\sigma_a$ to $a \\bmod n$, with $\\sigma_a(\\zeta_n) = \\zeta_n^a$. From: gal-jacobson Learn more:

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📐 Existence of the Algebraic Closure Every field $F$ has an algebraic closure $\\overline{F}$, and any two algebraic closures of $F$ are isomorphic over $F$. From: gal-weintraub Learn more:

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📐 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 Learn more:

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📐 Jordan Normal Form Every operator on a finite-dimensional complex vector space has a Jordan normal form: a block diagonal matrix with Jordan blocks $\\begin{pmatrix} \\lambda & 1 \\\\ & \\lambda & \\ddots \\\\ & & \\ddots & 1 \\\\ & & & \\lambda \\end{pmatrix}$. From: linalg-axler Learn more:

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📖 Characteristic of a Field The characteristic of a field $F$, denoted $\\operatorname{char}(F)$, is the smallest positive integer $p$ such that $p \\cdot 1 = 0$, or $0$ if no such integer exists. From: gal-weintraub Learn more:

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📖 Complexification The complexification of a real vector space $V$ is $V_C = V \\times V$ with $(u_1, u_2) + (v_1, v_2) = (u_1 + v_1, u_2 + v_2)$ and $(a + bi)(u, v) = (au - bv, av + bu)$. From: linalg-axler Learn more:

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📖 Cyclotomic Polynomial The $n$th **cyclotomic polynomial** is $\\Phi_n(x) = \\prod (x - \\zeta)$ where the product ranges over all primitive $n$th roots of unity. We have $x^n - 1 = \\prod_{d | n} \\Phi_d(x)$ and $\\deg \\Phi_n = \\varphi(n)$. From: gal-morandi Learn more:

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📖 Field Generated by Adjunction The set $F(\\alpha, \\beta, \\gamma, \\ldots)$ is the smallest extension of $F$ containing the elements $\\alpha, \\beta, \\gamma, \\ldots$. It consists of all quotients of polynomials in these elements with coefficients in $F$. From: gal-artin Learn more:

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📖 Natural Irrationalities Given a splitting field $E$ of $p(x)$ over $F$ and an arbitrary extension $B$ of $F$, the roots of $p(x)$ that generate $E$ over $F$ are the natural irrationalities. They live simultaneously in $E$ and in $EB$ (the splitting field over $B$). From: gal-artin Learn more:

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🎮 Interactive: Eulerian Path Finder Find paths that traverse every edge exactly once. Euler solved this for the Seven Bridges of Konigsberg in 1736! From: Introduction to Graph Theory Try it:

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🎮 Interactive: Homogeneous Coordinates Demo Learn how adding an extra coordinate lets us handle points at infinity. The key trick that makes projective geometry work. From: Four Pillars of Geometry Try it:

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📐 Discriminant Test Let $f(x) \\in K[x]$ be separable of degree $n$ with Galois group $G \\leq S_n$. Then $G \\subseteq A_n$ if and only if the discriminant $\\Delta(f)$ is a perfect square in $K$. From: gal-jacobson Learn more:

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📐 Separability via Differentials A finitely generated extension $K/F$ is separable if and only if $\\dim_K \\Omega_{K/F} = \\operatorname{tr.deg}(K/F)$. For an algebraic extension, separability is equivalent to $\\Omega_{K/F} = 0$. From: gal-morandi Learn more:

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📖 Constructible Number A real number $\\alpha$ is **constructible** if it can be obtained from $0$ and $1$ by a finite sequence of ruler and compass operations. The set of constructible numbers forms a field. From: gal-morandi Learn more:

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📐 Structure of Finite Fields For every prime $p$ and positive integer $n$, there exists a unique (up to isomorphism) field $\\mathbb{F}_{p^n}$ with $p^n$ elements. Its multiplicative group is cyclic of order $p^n - 1$. From: gal-weintraub Learn more:

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📖 Group (in Galois A set of substitutions of $n$ objects $a, b, c, \\ldots$ is called a group if the composition of any two substitutions in the set is again in the set. In modern terms, a finite set of permutations that is closed under composition is automatically a group (the identity and inverses come for free in the finite case). From: gal-edwards Learn more:

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📖 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 Learn more:

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📖 Axiom 3: Normalization The determinant is $1$ when the columns are the standard unit vectors $U_1, U_2, \\ldots, U_n$. From: gal-artin Learn more:

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