📖 Simple Root of $F(X)$ The polynomial $t$ in the roots is a simple root of its equation $F(X)$ if the polynomials $t, \\phi_1 t, \\phi_2 t, \\ldots$ obtained by distinct permutations of the roots are all numerically distinct (when actual root values are substituted). In modern language, this means $F(X)$ has no repeated roots. From: gal-edwards Learn more:

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📖 Reducible and Irreducible Polynomials A polynomial in $F$ is reducible if it equals the product of two polynomials in $F$ each of degree at least one. Polynomials that are not reducible are called irreducible. From: gal-artin Learn more:

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📖 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:

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📐 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 Learn more:

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📐 Genus and the Riemann-Roch Space For an algebraic function field $K/k$, the **genus** $g$ is defined such that for any divisor $D$, the Riemann-Roch space $L(D)$ is a finite-dimensional $k$-vector space with $\\dim L(D) \\geq \\deg D - g + 1$. From: gal-morandi Learn more:

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📐 Characterization of Normal Subgroups A subgroup $H$ of a group $G$ is normal if and only if for every $S \\in H$ and every $T \\in G$, the element $T^{-1}ST$ is in $H$. Proof: A subgroup $H$ has the property that every presentation differs from the first by a single substitution if and only if for every $T$ in $G$ and every $S$ in $H$, $T^{-1}ST$ is in $H$. In other words, the subgroup is invariant under conjugation by elements of $G$. From: gal-edwards Learn more:

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📐 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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