🔗 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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📐 Fundamental Identity for Valuations Let $L/K$ be a finite extension of degree $n$ and $v$ a valuation on $K$. If $v_1, \\ldots, v_g$ are the extensions of $v$ to $L$ with ramification indices $e_i$ and residue degrees $f_i$, then $\\sum_{i=1}^{g} e_i f_i = n$. From: gal-jacobson Learn more:

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📐 Theorem 5 A system of $n$ non-homogeneous linear equations in $n$ unknowns has a unique solution if and only if the corresponding homogeneous system has only the trivial solution. Proof: If the non-homogeneous system has two solutions, their difference solves the homogeneous system non-trivially. Conversely, if the homogeneous system has only the trivial solution, the column vectors are independent and form a generating system, so the non-homogeneous system has a unique solution. From: gal-artin Learn more:

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📖 Cyclotomic Polynomial For a prime $p$, the cyclotomic polynomial is $\\Phi_p(x) = \\frac{x^p - 1}{x - 1} = x^{p-1} + x^{p-2} + \\cdots + x + 1$. This polynomial is irreducible over $\\mathbb{Q}$. Its roots are the primitive $p$th roots of unity. From: gal-edwards Learn more:

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🎮 Interactive: Non-Euclidean Lines Demo Explore straight lines in hyperbolic geometry. In the Poincare disk, geodesics appear as circular arcs perpendicular to the boundary. From: Four Pillars of Geometry Try it:

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📖 Dual Space The dual space of $V$, denoted $V\ From: linalg-axler Learn more:

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📖 Normal Basis For a normal extension $E/F$ with Galois group $G = \\{\\sigma_1, \\ldots, \\sigma_n\\}$, a normal basis is an element $\\theta \\in E$ such that $\\sigma_1(\\theta), \\ldots, \\sigma_n(\\theta)$ are linearly independent over $F$ (and hence form a basis). From: gal-artin Learn more:

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📐 Galois Group of the Cyclotomic Equation The Galois group of $x^p - 1 = 0$ over $\\mathbb{Q}$ (equivalently, the Galois group of $\\Phi_p(x) = 0$ over $\\mathbb{Q}$) is the cyclic group of order $p - 1$, isomorphic to $(\\mathbb{Z}/p\\mathbb{Z})^*$. Each automorphism sends $a$ to some power $a^j$ where $j \\not\\equiv 0 \\pmod{p}$. From: gal-edwards Learn more:

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📖 Primitive Root Modulo p For a prime $p$, an integer $g$ is called a primitive root modulo $p$ if every integer not divisible by $p$ is congruent to a power of $g$ modulo $p$. Equivalently, the powers $g, g^2, g^3, \\ldots, g^{p-1}$ give all $p - 1$ nonzero residue classes modulo $p$. From: gal-edwards Learn more:

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📐 Ostrowski Every nontrivial absolute value on $\\mathbb{Q}$ is equivalent to either the usual absolute value $|\\cdot|_\\infty$ or the $p$-adic absolute value $|\\cdot|_p$ for some prime $p$. From: gal-jacobson Learn more:

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📐 Hilbert Theorem 90 Let $K/F$ be a cyclic Galois extension with generator $\\sigma$ of $\\operatorname{Gal}(K/F)$. An element $\\beta \\in K$ has $N_{K/F}(\\beta) = 1$ if and only if $\\beta = \\alpha / \\sigma(\\alpha)$ for some $\\alpha \\in K^\\times$. From: gal-morandi Learn more:

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📖 Subgroup A subgroup of a group $G$ is a group that is contained in $G$. The number of elements in the subgroup must divide the number of elements in the group, and the quotient is the index. This is Lagrange\ From: gal-edwards Learn more:

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📐 Existence of Splitting Fields Every polynomial $f \\in F[X]$ of degree $\\geq 1$ has a splitting field, and any two splitting fields of $f$ over $F$ are isomorphic via an isomorphism fixing $F$. From: gal-weintraub Learn more:

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📐 Constructibility Criterion A length $\\alpha$ is constructible by straightedge and compass if and only if $\\alpha$ lies in a field extension of $\\mathbb{Q}$ of degree $2^n$ for some $n \\geq 0$. From: gal-weintraub Learn more:

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📐 Unique Factorization for Polynomials over $\\mathbb{Z}$ A representation of a polynomial with integer coefficients as a product of irreducibles is unique up to the order of the factors and their signs. That is, if $F_1 F_2 \\cdots F_\\mu = G_1 G_2 \\cdots G_\\nu$ where all factors are irreducible, then $\\mu = \\nu$ and the $G_j$\ From: gal-edwards Learn more:

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

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📐 Isomorphism Extension Theorem Let $\\sigma: F \\to F\ From: gal-weintraub Learn more:

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🎮 Interactive: Linear Fractional Transformation Demo Explore transformations of the form (ax+b)/(cx+d). These maps form a group and connect algebra to geometry beautifully. From: Four Pillars of Geometry Try it:

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📖 Sample Definition A function $f: A \\to B$ is a mapping from set $A$ to set $B$. From: gal-howie 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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