📖 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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📖 Primitive Extension An extension $E/F$ is primitive (simple) if $E = F(\\theta)$ for a single element $\\theta$, called a primitive element. From: gal-artin Learn more:

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📖 Kummer Extension If $F$ contains a primitive $n$th root of unity, a splitting field of $p(x) = (x^n - a_1)(x^n - a_2) \\cdots (x^n - a_r)$ with $a_i \\in F$ is a Kummer extension of $F$. From: gal-artin Learn more:

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📐 Theorem 4 The right column rank, left column rank, right row rank, and left row rank of a matrix are all equal. Proof: Show $c \\leq r$ by truncating to the first $r$ independent rows (the column rank does not change). Applying the same argument to the transpose gives $r \\leq c$, hence $r = c$. The same reasoning equates all four rank notions. From: gal-artin Learn more:

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📐 Uniqueness of Splitting Fields Let $f(x)$ be a polynomial with coefficients in a field $K$, and let $L_1$ and $L_2$ be two splitting fields for $f(x)$ over $K$. Then there is an isomorphism $\\sigma: L_1 \\to L_2$ that fixes every element of $K$. From: gal-edwards Learn more:

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📖 Vector Space A vector space over $\\mathbf{F}$ is a set $V$ with addition $V \\times V \\to V$ and scalar multiplication $\\mathbf{F} \\times V \\to V$ satisfying commutativity, associativity, additive identity, additive inverse, multiplicative identity, and distributive properties. From: linalg-axler Learn more:

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📐 Kummer Theory Let $K$ be a field containing a primitive $n$-th root of unity $\\zeta_n$ (with $\\mathrm{char}(K) \\nmid n$). There is a bijection between cyclic extensions $L/K$ of degree dividing $n$ and subgroups of $K^\\times/(K^\\times)^n$, given by $L = K(a^{1/n})$ for $a \\in K^\\times$. From: gal-jacobson Learn more:

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📖 Solving by Radicals An equation is said to be solvable by radicals if its roots can be expressed in terms of its coefficients using only the operations of addition, subtraction, multiplication, division, and the extraction of $n$th roots for various $n$. From: gal-edwards Learn more:

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📐 Absolute Galois Group of Finite Fields The absolute Galois group $\\operatorname{Gal}(\\overline{\\mathbb{F}_p}/\\mathbb{F}_p) \\cong \\hat{\\mathbb{Z}}$, the profinite completion of $\\mathbb{Z}$, topologically generated by the Frobenius $a \\mapsto a^p$. From: gal-morandi Learn more:

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📐 Singular Value Decomposition If $T \\in \\mathcal{L}(V)$, then there exist orthonormal bases $e_1, \\ldots, e_n$ and $f_1, \\ldots, f_n$ of $V$ such that $Tv = s_1\\langle v, e_1\\rangle f_1 + \\cdots + s_n\\langle v, e_n\\rangle f_n$ where $s_1 \\geq \\cdots \\geq s_n \\geq 0$ are the singular values of $T$. From: linalg-axler Learn more:

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📖 Direct Sum $V = U_1 \\oplus U_2$ means every $v \\in V$ can be written uniquely as $v = u_1 + u_2$ with $u_1 \\in U_1$, $u_2 \\in U_2$. Equivalently, $V = U_1 + U_2$ and $U_1 \\cap U_2 = \\{0\\}$. From: linalg-axler Learn more:

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🎮 Interactive: Classification Boundary Visualizer See how classifiers divide feature space into regions. Compare logistic regression, LDA, and KNN boundaries. From: Intro to Statistical Learning Try it:

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📖 Symmetric Polynomial A polynomial $g(x_1, \\ldots, x_n)$ is symmetric if it is unchanged under all permutations of its variables. Every symmetric polynomial can be expressed as a polynomial in the elementary symmetric functions $a_1, \\ldots, a_n$. From: gal-artin Learn more:

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📐 Frobenius Automorphism The Galois group $\\operatorname{Gal}(\\mathbb{F}_{p^n}/\\mathbb{F}_p)$ is cyclic of order $n$, generated by the Frobenius automorphism $\\phi: a \\mapsto a^p$. From: gal-morandi 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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📐 Separable Degree For a finite extension $K/F$, the number of $F$-homomorphisms from $K$ into an algebraic closure of $F$ equals the separable degree $[K:F]_s$, which divides $[K:F]$. From: gal-morandi Learn more:

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🎮 Interactive: Linear Regression Fitter Fit a line to data points using least squares. See how regression minimizes the sum of squared residuals. From: Intro to Statistical Learning Try it:

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📐 Multiplication Theorem for Determinants If $A$ and $B$ are $n \\times n$ matrices, then $|AB| = |A| \\cdot |B|$. Proof: Replace each column $A_k$ by $\\sum b_{\\nu k} A_\\nu$. Expanding using Axiom 1, only terms with distinct indices survive. The resulting sum equals $|A| \\cdot |B|$ by the Leibniz formula. From: gal-artin Learn more:

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📐 Basis Length Any two bases of a finite-dimensional vector space have the same length. From: linalg-axler Learn more:

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📐 Existence and Uniqueness of Splitting Fields For any nonconstant $f(x) \\in F[x]$, a splitting field exists and is unique up to $F$-isomorphism. From: gal-morandi Learn more:

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