📖 Parity check A parity check on strings of length $n$ is a linear equation $h_1 u_1 + h_2 u_2 + \\\\cdots + h_n u_n = 0$ with fixed coefficients $h_i$ in GF(32). A code defined as the set of strings satisfying a fixed collection of parity checks is automatically linear, since the solution set of a system of homogeneous linear equations is a subspace. From: codex32 Learn more:

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📖 Detection, correction, and erasure filling Detection is answering "this string is not valid". It requires no knowledge of what the right string was. Correction is answering "the right string was this one", when neither the number of damaged positions nor their locations is known in advance. Erasure filling is answering "the missing symbols were these", when the damaged positions are known and only their contents are unknown. From: codex32 Learn more:

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📖 A validated share set A collection of shares carrying evidence that: the threshold is nonzero and identical across all of them; the identifiers match; the payload lengths match; the share indices are pairwise distinct; and none of them is the secret index. From: codex32 Learn more:

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📐 Nonzero elements of GF(32) are invertible For every nonzero $a \\\\in \\\\mathrm{GF}(32)$ there is exactly one $b$ with $a b = 1$. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📖 Generic codex32 A prefix, a six-symbol header, a payload of anywhere from 0 to 997 symbols, and a checksum of the size the length rules demand. Nothing constrains what the payload means. From: codex32 Learn more:

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📖 Zero divisor A nonzero element $a$ is a zero divisor if there is some nonzero $b$ with $a \\\\times b = 0$. From: codex32 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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📖 Solvable Group (Preview) A group $G$ is solvable if there exists a chain of subgroups $\\{e\\} = G_0 \\triangleleft G_1 \\triangleleft G_2 \\triangleleft \\cdots \\triangleleft G_k = G$ where each $G_i$ is normal in $G_{i+1}$ and each quotient $G_{i+1}/G_i$ is cyclic of prime order. $S_3$ and $S_4$ are solvable, but $S_5$ is not. From: gal-edwards Learn more:

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📐 Hilbert Theorem 90 (Additive Form) Let $K/F$ be a cyclic Galois extension with generator $\\sigma$. An element $\\beta \\in K$ has $T_{K/F}(\\beta) = 0$ if and only if $\\beta = \\alpha - \\sigma(\\alpha)$ for some $\\alpha \\in K$. From: gal-morandi 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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📐 Cramer If the determinant $D$ of the coefficient matrix is nonzero, the system $\\sum_j a_{ij}x_j = b_i$ has the unique solution $D \\cdot x_k = \\sum_i A_{ik} b_i$, where $A_{ik}$ are the cofactors. Proof: Multiply the $i$-th equation by $A_{ik}$ and sum over $i$. By the orthogonality relations of cofactors, $\\sum_i a_{ij} A_{ik} = D$ if $j=k$ and $0$ otherwise. This gives $D \\cdot x_k = \\sum_i A_{ik} b_i$. From: gal-artin Learn more:

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📐 Dimension of a Sum If $U_1$ and $U_2$ are subspaces of a finite-dimensional vector space, then $\\dim(U_1 + U_2) = \\dim U_1 + \\dim U_2 - \\dim(U_1 \\cap U_2)$. From: linalg-axler Learn more:

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📖 Cyclotomic Polynomial The $n$th **cyclotomic polynomial** is $\\Phi_n(x) = \\prod (x - \\zeta)$ where the product ranges over all primitive $n$th roots of unity. We have $x^n - 1 = \\prod_{d | n} \\Phi_d(x)$ and $\\deg \\Phi_n = \\varphi(n)$. From: gal-morandi 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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📖 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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📖 Cyclotomic Equation The equation $x^n = 1$, or equivalently $x^n - 1 = 0$, is called the cyclotomic equation. Its solutions are the $n$th roots of unity. The name comes from the Greek words for "circle" and "division," because the roots divide a circle into $n$ equal parts. From: gal-edwards Learn more:

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📐 Newton\ The power sums $s_k = r_1^k + r_2^k + \\cdots + r_n^k$ satisfy the recurrence relation: $s_k - s_{k-1}\\sigma_1 + s_{k-2}\\sigma_2 - \\cdots + (-1)^{k-1}s_1\\sigma_{k-1} + (-1)^k k\\sigma_k = 0$ for $k = 1, 2, 3, \\ldots$, where $\\sigma_j = 0$ for $j > n$. Proof: This recurrence follows from the identity $r_i^n - \\sigma_1 r_i^{n-1} + \\sigma_2 r_i^{n-2} - \\cdots \\pm \\sigma_n = 0$, which holds for each root $r_i$. Summing over $i$ and using the definition of the power sums gives $s_n - \\sigma_1 s_{n-1} + \\sigma_2 s_{n-2} - \\cdots \\pm n\\sigma_n = 0... From: gal-edwards 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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📐 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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