📖 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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📐 Galois A polynomial $f(x) \\in K[x]$ (with $\\mathrm{char}(K) = 0$) is solvable by radicals if and only if its Galois group $\\mathrm{Gal}(f)$ is a solvable group. From: gal-jacobson Learn more:

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📖 Irreducible Element (in a Ring) An element $p$ of a ring is irreducible if it is not a unit (not $\\pm 1$ in $\\mathbb{Z}$) and if the only factorizations $p = ab$ have one of $a$, $b$ as a unit. In $\\mathbb{Z}$, irreducible elements are the prime numbers and their negatives. From: gal-edwards Learn more:

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📐 Theorem 6 (Tower Law) If $F \\subset B \\subset E$ are three fields, then $(E/F) = (B/F)(E/B)$. Proof: If $A_1, \\ldots, A_r$ are independent over $B$ and $C_1, \\ldots, C_s$ are independent over $F$, then the $rs$ products $C_i A_j$ are independent over $F$. A dependence relation $\\sum a_{ij} C_i A_j = 0$ forces $\\sum a_{ij} C_i = 0$ for each $j$ (by independence of $A_j$ over $B$), which force... From: gal-artin Learn more:

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📐 Generalized Eigenspace Decomposition If $V$ is a complex vector space and $T \\in \\mathcal{L}(V)$ has eigenvalues $\\lambda_1, \\ldots, \\lambda_m$, then $V = G(\\lambda_1, T) \\oplus \\cdots \\oplus G(\\lambda_m, T)$. From: linalg-axler Learn more:

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📐 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 Learn more:

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📖 Divisibility of Polynomials If $f(x) = g(x) \\cdot h(x)$, then $g(x)$ divides $f(x)$ in $F$, or equivalently, $g(x)$ is a factor of $f(x)$. From: gal-artin Learn more:

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📐 Structure of Simple Algebraic Extensions If $\\alpha$ is algebraic over $K$ with minimal polynomial $f(x)$ of degree $n$, then $K(\\alpha) \\cong K[x]/(f(x))$ and $[K(\\alpha):K] = n$. A basis for $K(\\alpha)/K$ is $\\{1, \\alpha, \\alpha^2, \\ldots, \\alpha^{n-1}\\}$. From: gal-jacobson Learn more:

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📖 Inner Product An inner product on $V$ is a function $\\langle \\cdot, \\cdot \\rangle: V \\times V \\to \\mathbf{F}$ satisfying positivity, definiteness, additivity in the first slot, homogeneity in the first slot, and conjugate symmetry. From: linalg-axler Learn more:

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📖 Solvable Group A group $G$ is said to be solvable if there exists a sequence of subgroups $G = G_0 \\supset G_1 \\supset G_2 \\supset \\cdots \\supset G_\\nu = \\{e\\}$ in which each $G_i$ is a normal subgroup of $G_{i-1}$ of prime index, and the final subgroup $G_\\nu$ contains only the identity. Such a sequence is called a composition series (with prime factors). From: gal-edwards Learn more:

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📖 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 Learn more:

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📐 L\u00FCroth If $K \\subset L \\subset K(x)$ with $L \\neq K$ (where $x$ is transcendental over $K$), then $L = K(u)$ for some $u = f(x)/g(x) \\in K(x)$. Every intermediate field of a simple transcendental extension is itself simple transcendental. From: gal-jacobson Learn more:

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📐 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 Learn more:

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