🎯 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 Learn more:

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📐 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 Learn more:

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📐 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 Learn more:

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📐 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 Learn more:

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📖 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 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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📖 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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📖 Field A field $F$ is a commutative ring with identity $1 \\neq 0$ in which every nonzero element has a multiplicative inverse. From: gal-weintraub Learn more:

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📐 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:

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📖 Generalized Eigenvector A vector $v \\in V$ is a generalized eigenvector of $T$ corresponding to $\\lambda$ if $v \\neq 0$ and $(T - \\lambda I)^j v = 0$ for some positive integer $j$. From: linalg-axler Learn more:

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📐 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 Learn more:

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📖 Primitive Root of Unity An element $\\epsilon$ is a primitive $n$th root of unity if $\\epsilon^n = 1$ and $\\epsilon$ has order exactly $n$. The roots of $x^n - 1$ are $1, \\epsilon, \\epsilon^2, \\ldots, \\epsilon^{n-1}$. From: gal-artin Learn more:

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🎯 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 Learn more:

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📐 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 Learn more:

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📐 Theorem 19 (Artin) A polynomial $f$ has repeated roots if and only if $f$ and $f\ From: gal-artin Learn more:

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📐 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 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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📖 Splitting Field A **splitting field** for a polynomial $f(x) \\in F[x]$ is an extension $K/F$ such that $f$ splits completely in $K[x]$ and $K = F(\\alpha_1, \\ldots, \\alpha_n)$ where the $\\alpha_i$ are the roots of $f$. From: gal-morandi Learn more:

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📖 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 Learn more:

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🎮 Interactive: RSA Encryption Demo See how RSA public-key encryption works. The security relies on the difficulty of factoring large numbers! From: Cryptography Math Try it:

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