📐 Galois Groups of Cubics An irreducible cubic $f(x) \\in \\mathbb{Q}[x]$ has Galois group $S_3$ if $\\operatorname{disc}(f)$ is not a perfect square in $\\mathbb{Q}$, and $A_3 \\cong \\mathbb{Z}/3\\mathbb{Z}$ if it is. From: gal-morandi Learn more:

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📖 Basis A basis of $V$ is a list of vectors in $V$ that is linearly independent and spans $V$. From: linalg-axler Learn more:

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📐 Structure of Roots of Unity To find primitive $n$th roots of unity for an arbitrary positive integer $n$, it suffices to be able to find primitive $p$th roots of unity for each prime $p$. This is because any $n$ can be written as a product of prime powers, and the $p^{n-1}$st root of a primitive $p$th root of unity is a primitive $p^n$th root. Proof: When $n = jk$ where $j$ and $k$ are relatively prime, the product of a primitive $j$th root and a primitive $k$th root is a primitive $n$th root. More generally, if $n = j \\cdot k \\cdots m$ with pairwise coprime factors, the primitive $n$th roots of unity are products of primitive roots of the ... From: gal-edwards Learn more:

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📖 Character of a Group A homomorphism $\\sigma$ from a multiplicative group $G$ into a field $F$ (with $\\sigma(\\alpha) \\neq 0$ for all $\\alpha$) is called a character of $G$ in $F$. From: gal-artin Learn more:

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📐 Galois If an element of the splitting field $K(a, b, c, \\ldots)$ is left fixed by all the automorphisms of the Galois group, then it is in $K$. Equivalently: the elements of $K$ are precisely those elements of the splitting field that are fixed by every element of the Galois group. From: gal-edwards Learn more:

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🔗 Lemma 2 (Maximal Order Divides All) If $C$ has maximal order $c$ in an abelian group, then $c$ is divisible by the order of every element, so $x^c = 1$ for all elements. From: gal-artin Learn more:

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📖 Intermediate Fields $K_d$ For each divisor $d$ of $p - 1$, let $K_d$ denote the set of all elements of $\\mathbb{Q}(\\alpha)$ invariant under $S^d$ (replacing $\\alpha$ by $\\alpha^{g^d}$). Then $K_1 = \\mathbb{Q}$ and $K_{p-1} = \\mathbb{Q}(\\alpha)$. The intermediate fields form a chain $\\mathbb{Q} = K_1 \\subset K_d \\subset K_D \\subset K_{p-1} = \\mathbb{Q}(\\alpha)$ whenever $d | D | (p-1)$. From: gal-edwards Learn more:

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📐 Theorem 14 (Artin) If $\\sigma_1, \\ldots, \\sigma_n$ form a group of automorphisms of $E$ with fixed field $F$, then $(E/F) = n$. From: gal-artin 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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📐 Fundamental Theorem of Algebra Every nonconstant polynomial with complex coefficients has a zero in $\\mathbf{C}$. From: linalg-axler Learn more:

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📖 Primitive Root of Unity An $n$th root of unity is a number $\\alpha$ satisfying $\\alpha^n = 1$. It is primitive if no smaller positive power of $\\alpha$ equals 1. For $n = 3$, the primitive cube roots of unity are $\\alpha = (-1 \\pm \\sqrt{-3})/2$. From: gal-edwards Learn more:

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📖 Algebraic Element An element $\\alpha \\in E$ is algebraic over $K$ if there exists a nonzero polynomial $f(x) \\in K[x]$ with $f(\\alpha) = 0$. Otherwise $\\alpha$ is transcendental over $K$. From: gal-jacobson Learn more:

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🎮 Interactive: Eigenvalue Explorer Find vectors that only get scaled by a matrix. Eigenvalues reveal the fundamental structure of linear transformations. From: Linear Algebra Try it:

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📖 Real Closed Field A field $R$ is real closed if it is ordered and has no proper algebraic extension that can be ordered. Equivalently, $R$ is real closed iff $R(\\sqrt{-1})$ is algebraically closed. From: gal-jacobson Learn more:

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📖 Left Vector Space An additive abelian group $V$ over a field $F$ with scalar multiplication $aA$ satisfying: (1) $a(A+B) = aA + aB$, (2) $(a+b)A = aA + bA$, (3) $a(bA) = (ab)A$, (4) $1A = A$. From: gal-artin Learn more:

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📐 Galois Group of Cyclotomic Extensions The cyclotomic polynomial $\\Phi_n(x)$ is irreducible over $\\mathbb{Q}$. The Galois group $\\operatorname{Gal}(\\mathbb{Q}(\\zeta_n)/\\mathbb{Q}) \\cong (\\mathbb{Z}/n\\mathbb{Z})^\\times$, sending $\\sigma_a: \\zeta_n \\mapsto \\zeta_n^a$ to $a \\bmod n$. From: gal-morandi Learn more:

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📖 Right Vector Space An additive abelian group $V$ over a field $F$ with scalar multiplication written on the right, $Aa$, satisfying: (1) $(A+B)a = Aa + Ba$, (2) $A(a+b) = Aa + Ab$, (3) $(Ab)a = A(ba)$, (4) $A1 = A$. 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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📐 Fundamental Theorem of Galois Theory Let $K/F$ be a finite Galois extension with Galois group $G = \\operatorname{Gal}(K/F)$. There is an inclusion-reversing bijection between intermediate fields $F \\subseteq L \\subseteq K$ and subgroups $H \\leq G$, given by $L \\mapsto \\operatorname{Gal}(K/L)$ and $H \\mapsto K^H$. Moreover, $[K:L] = |H|$ and $[L:F] = [G:H]$, and $L/F$ is normal if and only if $H$ is normal in $G$. From: gal-morandi Learn more:

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📖 Separable Polynomial and Separable Extension A polynomial is separable if its irreducible factors have no repeated roots. An element is separable if it is a root of a separable polynomial. The extension $E/F$ is separable if every element of $E$ is separable over $F$. From: gal-artin Learn more:

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