📖 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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📖 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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🔗 Lemma (Irreducible Divisibility) If $f(x)$ is irreducible of degree $n$, there do not exist two polynomials each of degree less than $n$ whose product is divisible by $f(x)$. Proof: Suppose $g(x)h(x)$ is divisible by $f(x)$ with $\\deg(g), \\deg(h) < n$. Choose $g$ of minimal degree. Dividing $f$ by $g$ gives $f = qg + r$ with $0 < \\deg(r) < \\deg(g)$. Then $r \\cdot h$ is divisible by $f$, contradicting the minimality of $\\deg(g)$. From: gal-artin Learn more:

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