📐 Fundamental Theorem on Symmetric Functions Every symmetric polynomial in $x_1, \\ldots, x_n$ over a field $F$ can be uniquely expressed as a polynomial in the elementary symmetric polynomials $e_1, \\ldots, e_n$. From: gal-weintraub Learn more:

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📖 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:

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📐 Decomposition Theorem Each abelian group with finitely many generators is the direct product of cyclic subgroups $G_1, \\ldots, G_n$ where the order of $G_i$ divides the order of $G_{i+1}$. From: gal-artin Learn more:

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📖 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:

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🔗 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 Learn more:

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📖 Subspace A subset $U$ of $V$ is a subspace of $V$ if $U$ is also a vector space with the same operations. Equivalently: $0 \\in U$, $U$ is closed under addition, and $U$ is closed under scalar multiplication. From: linalg-axler Learn more:

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📖 Cycle Notation The mapping $T(i) = j, T(j) = k, \\ldots, T(m) = i$ is written $(i\\,j\\,\\ldots\\,m)$ and called a $k$-cycle. A 2-cycle $(i\\,j)$ is a transposition. From: gal-artin Learn more:

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🔗 Lemma (Irreducible Divisibility) If $f(x)$ is irreducible of degree $n$, there do not exist two polynomials each of degree less than $n$ whose product is divisible by $f(x)$. Proof: Suppose $g(x)h(x)$ is divisible by $f(x)$ with $\\deg(g), \\deg(h) < n$. Choose $g$ of minimal degree. Dividing $f$ by $g$ gives $f = qg + r$ with $0 < \\deg(r) < \\deg(g)$. Then $r \\cdot h$ is divisible by $f$, contradicting the minimality of $\\deg(g)$. From: gal-artin Learn more:

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📐 Cyclotomic Extension The $n$-th cyclotomic polynomial $\\Phi_n(X)$ is irreducible over $\\mathbb{Q}$, of degree $\\varphi(n)$. The Galois group $\\operatorname{Gal}(\\mathbb{Q}(\\zeta_n)/\\mathbb{Q}) \\cong (\\mathbb{Z}/n\\mathbb{Z})^\\times$. From: gal-weintraub Learn more:

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📖 Galois Extension A finite extension $E/K$ is Galois if it is both normal and separable. Equivalently, $|\\mathrm{Aut}(E/K)| = [E:K]$. The Galois group is $\\mathrm{Gal}(E/K) = \\mathrm{Aut}(E/K)$. From: gal-jacobson Learn more:

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📐 Existence and Uniqueness of Finite Fields For every prime power $q = p^n$, there exists a unique (up to isomorphism) field $\\mathbb{F}_q$ with $q$ elements. Its multiplicative group $\\mathbb{F}_q^\\times$ is cyclic of order $q - 1$. From: gal-morandi Learn more:

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📐 Theorem 13 (Artin) If $\\sigma_1, \\ldots, \\sigma_n$ are $n$ mutually distinct isomorphisms of $E$ into $E\ From: gal-artin Learn more:

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📖 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 Learn more:

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📖 The Groups A, F_1, and Their Powers Let $A$ be the set of nonzero $a \\in E$ with $a^r \\in F$ (where $r = \\mathrm{lcm}$ of orders in $G$), $F_1$ the nonzero elements of $F$. Then $A$ and $F_1$ are multiplicative groups, and $A^s$, $F_1^s$ denote their sets of $s$th powers. From: gal-artin Learn more:

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📖 Linear Dependence Vectors $A_1, \\ldots, A_n$ in a vector space $V$ over $F$ are dependent if there exist scalars $x_1, \\ldots, x_n \\in F$, not all zero, such that $x_1 A_1 + \\cdots + x_n A_n = 0$. From: gal-artin Learn more:

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📐 Theorem 17 (Artin) If $S$ is a finite subset ($\\neq 0$) of a field $F$ which is a group under multiplication, then $S$ is a cyclic group. From: gal-artin Learn more:

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📖 Separable Polynomial and Extension A polynomial $f(x) \\in F[x]$ is **separable** if it has no repeated roots in any extension. An algebraic extension $K/F$ is **separable** if the minimal polynomial of every element of $K$ over $F$ is separable. From: gal-morandi Learn more:

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📖 Separable Polynomial A polynomial $f \\in F[X]$ is separable if it has no repeated roots in any extension of $F$. An algebraic element is separable if its minimal polynomial is separable. From: gal-weintraub Learn more:

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📖 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 Learn more:

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📖 Linear Independence A list $v_1, \\ldots, v_m$ in $V$ is linearly independent if the only choice of $a_1, \\ldots, a_m \\in \\mathbf{F}$ that makes $a_1 v_1 + \\cdots + a_m v_m = 0$ is $a_1 = \\cdots = a_m = 0$. From: linalg-axler Learn more:

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