🎮 Interactive: Isometry Group Demo Discover the isometries of 3D space: translations, rotations, reflections, and glide reflections. Every distance-preserving map is one of these! From: Four Pillars of Geometry Try it:

The Four Pillars of Geometry | Magic Internet Math

Explore geometry from four perspectives: Euclidean constructions, coordinates, projective geometry, and transformations. Based on John Stillwell

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📐 Trace is Basis-Independent $\\operatorname{trace} T = \\operatorname{trace} \\mathcal{M}(T)$ for any basis, where the trace of a matrix is the sum of its diagonal entries. From: linalg-axler Learn more:

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📐 Determinant of a Product $\\det(ST) = (\\det S)(\\det T)$ for operators $S, T \\in \\mathcal{L}(V)$. From: linalg-axler Learn more:

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📖 Solvable by Radicals A polynomial $f(x)$ in $F$ is solvable by radicals if its splitting field lies in an extension of $F$ by radicals. From: gal-artin 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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📖 Formal Derivative For $f(x) = a_0 + a_1 x + \\cdots + a_n x^n$, the formal derivative is $f\ From: gal-artin Learn more:

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📖 Lagrange Resolvent (for the Cubic) For the cubic with roots $x, y, z$, the Lagrange resolvent is the quantity $t = x + \\alpha y + \\alpha^2 z$ where $\\alpha$ is a cube root of unity ($\\alpha \\neq 1$). The quantity $t$ has six values depending on the order of the roots, and these are the solutions of a 6th degree resolvent equation. From: gal-edwards Learn more:

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📖 Solvable by Radicals A polynomial $f(x) \\in F[x]$ is **solvable by radicals** if there is a tower $F = F_0 \\subset F_1 \\subset \\cdots \\subset F_n$ with $F_n$ containing all roots of $f$, where each $F_{i+1} = F_i(\\alpha_i)$ with $\\alpha_i^{n_i} \\in F_i$ for some $n_i$. From: gal-morandi Learn more:

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📐 Theorem 1: Existence of Factorization Every polynomial with integer coefficients that is not a unit (not $\\pm 1$) can be written as a product of irreducible polynomials with integer coefficients. From: gal-edwards Learn more:

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📐 Galois Criterion for Solvability A polynomial $f \\in F[X]$ is solvable by radicals if and only if its Galois group $\\operatorname{Gal}(E/F)$ is a solvable group. From: gal-weintraub Learn more:

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📖 Affine Group mod $p$ The affine group modulo a prime $p$ consists of all permutations of $\\{0, 1, \\ldots, p-1\\}$ of the form $j \\mapsto rj + s \\pmod{p}$ where $r \\not\\equiv 0 \\pmod{p}$ and $s \\in \\{0, 1, \\ldots, p-1\\}$. This group has order $p(p-1)$ and is solvable. Its normal subgroup of translations ($r = 1$) is cyclic of order $p$, and the quotient is cyclic of order $p - 1$. From: gal-edwards 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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📖 Transitive Action A group $G$ acts transitively on a set if for any two elements $a$ and $b$ of the set, there is an element of $G$ that maps $a$ to $b$. A polynomial $f(x)$ with no repeated roots is irreducible over $K$ if and only if the Galois group acts transitively on the roots. From: gal-edwards Learn more:

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📖 Galois Extension A finite extension $K/F$ is **Galois** if $|\\operatorname{Aut}(K/F)| = [K:F]$. Equivalently, $K/F$ is both normal and separable. From: gal-morandi Learn more:

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📐 Fundamental Theorem (Infinite Case) Let $K/F$ be a (possibly infinite) Galois extension. The Galois correspondence gives an inclusion-reversing bijection between intermediate fields and **closed** subgroups of $\\operatorname{Gal}(K/F)$ (in the Krull topology). An intermediate extension $L/F$ is Galois if and only if the corresponding subgroup is normal and closed. From: gal-morandi Learn more:

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📐 Characterization of Formally Real Fields A field $K$ admits an ordering if and only if $-1$ is not a sum of squares in $K$. Such fields are called formally real. From: gal-jacobson Learn more:

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📐 Cayley-Hamilton Theorem If $T \\in \\mathcal{L}(V)$ and $q$ is the characteristic polynomial of $T$, then $q(T) = 0$. From: linalg-axler Learn more:

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📖 Linear Independence Vectors $A_1, \\ldots, A_n$ are independent if the only choice of scalars for which $x_1 A_1 + \\cdots + x_n A_n = 0$ is the trivial one $x_1 = \\cdots = x_n = 0$. From: gal-artin Learn more:

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📖 Extension Field If $E$ is a field and $F$ is a subset of $E$ which itself forms a field under the operations of $E$, then $F$ is a subfield of $E$ and $E$ is an extension of $F$. From: gal-artin Learn more:

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📖 Purely Inseparable Extension An algebraic extension $K/F$ is **purely inseparable** if, for every $\\alpha \\in K$, the minimal polynomial of $\\alpha$ over $F$ has only one distinct root. In characteristic $p$, this means $\\alpha^{p^n} \\in F$ for some $n$. From: gal-morandi Learn more:

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