📖 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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📐 Existence of Galois Resolvents For any polynomial equation of degree $n$ with distinct roots, there exist integers $A, B, C, \\ldots$ such that $t = Aa + Bb + Cc + \\cdots$ has $n!$ distinct values under all $n!$ permutations of the roots. From: gal-edwards Learn more:

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🎮 Interactive: Pell Equation Solver Find integer solutions to x^2 - Dy^2 = 1. Watch continued fractions reveal the fundamental solution to these ancient Diophantine equations. From: Disquisitiones Arithmeticae Try it:

Disquisitiones Arithmeticae | Math Academy

Gauss

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📖 Discriminant The discriminant of a polynomial with roots $r_1, r_2, \\ldots, r_n$ is the product $D = \\prod_{1 \\leq i < j \\leq n} (r_i - r_j)^2$. The discriminant is symmetric in the roots and can be expressed in terms of the coefficients by the Fundamental Theorem. From: gal-edwards Learn more:

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📐 Invertibility Characterization A linear map $T \\in \\mathcal{L}(V, W)$ is invertible if and only if $T$ is injective and surjective. From: linalg-axler Learn more:

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📖 Irreducible Polynomial A polynomial $f \\in F[X]$ of degree $\\geq 1$ is irreducible over $F$ if it cannot be written as a product $f = gh$ with $\\deg(g), \\deg(h) \\geq 1$. From: gal-weintraub Learn more:

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📐 Tower Law If $F \\subseteq K \\subseteq L$ are fields, then $[L:F] = [L:K][K:F]$. Proof: If $\\{a_i\\}$ is a basis for $K/F$ and $\\{b_j\\}$ is a basis for $L/K$, then $\\{a_i b_j\\}$ is a basis for $L/F$. The result follows by counting. From: gal-morandi Learn more:

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📖 Constructible Number A real number $x$ is constructible (from given data $a, b, c, \\ldots$) if it can be obtained using a finite sequence of addition, subtraction, multiplication, division, and extraction of square roots. From: gal-artin Learn more:

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📐 Discriminant and Alternating Group Let $f(x) \\in F[x]$ be separable of degree $n$ with Galois group $G \\leq S_n$. Then $G \\subseteq A_n$ if and only if $\\operatorname{disc}(f)$ is a square in $F$. From: gal-morandi Learn more:

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📐 Isomorphism and Dimension Two finite-dimensional vector spaces over $\\mathbf{F}$ are isomorphic if and only if they have the same dimension. From: linalg-axler Learn more:

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📖 Minimal Polynomial The minimal polynomial of $\\alpha$ over $K$ is the unique monic irreducible polynomial $\\mathrm{min}_K(\\alpha) \\in K[x]$ such that $\\mathrm{min}_K(\\alpha)(\\alpha) = 0$. It divides every polynomial in $K[x]$ that has $\\alpha$ as a root. From: gal-jacobson Learn more:

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📐 Sample Theorem If $A \\subseteq B$ and $B \\subseteq A$, then $A = B$ Proof: Let $x \\in A$. Since $A \\subseteq B$, we have $x \\in B$ by definition of subset. Therefore, every element of $A$ is in $B$. Now, let $y \\in B$. Since $B \\subseteq A$, we have $y \\in A$ by definition. Therefore, every element of $B$ is in $A$. Since $A \\subseteq B$ and $B \\subseteq A... From: gal-howie Learn more:

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🎮 Interactive: Cross-Ratio Explorer Discover the cross-ratio, the fundamental invariant of projective geometry. See how this quantity stays constant under perspective transformations. 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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📐 Determinant and Invertibility An operator $T$ is invertible if and only if $\\det T \\neq 0$. From: linalg-axler Learn more:

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📖 Constructible Number A real number is constructible (by ruler and compass) if it can be obtained from the rational numbers by a finite sequence of operations involving addition, subtraction, multiplication, division, and the extraction of square roots. Equivalently, it belongs to a field that can be reached from $\\mathbb{Q}$ by a tower of quadratic extensions. From: gal-edwards Learn more:

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📖 Splitting Field An extension $E$ of $F$ in which $p(x)$ factors into linear factors, and $p(x)$ cannot be so factored in any intermediate field, is the splitting field of $p(x)$. Equivalently, $E$ is generated over $F$ by the roots of $p(x)$. From: gal-artin Learn more:

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📖 Dimension The dimension of a vector space $V$ over a field $F$ is the maximum number of independent elements in $V$. From: gal-artin Learn more:

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📖 Dimension The dimension of a finite-dimensional vector space is the length of any basis of the vector space, denoted $\\dim V$. From: linalg-axler Learn more:

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📐 Classical Fundamental Theorem of Algebra Every polynomial of degree $n \\geq 1$ with coefficients in $\\mathbb{C}$ has at least one root in $\\mathbb{C}$. Equivalently, every such polynomial can be written as a product of linear factors: $f(x) = c(x - z_1)(x - z_2) \\cdots (x - z_n)$ where $z_1, z_2, \\ldots, z_n \\in \\mathbb{C}$. From: gal-edwards Learn more:

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🔗 Lemma 1 (Existence of Primitive Roots) For every prime $p$ there exists an integer $g$ with the property that every integer not congruent to 0 modulo $p$ is congruent to a power of $g$ modulo $p$. Such an integer is called a primitive root modulo $p$. Proof: Gauss published the first rigorous proof in Article 55 of the Disquisitiones Arithmeticae. The proof considers the polynomial $x^d - 1 \\pmod{p}$ for each divisor $d$ of $p - 1$. A counting argument shows that for each divisor $d$ there are exactly $\\varphi(d)$ elements of order $d$. In particul... From: gal-edwards Learn more:

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