📐 Every order divides 1023 The order of any nonzero element of GF(1024) divides $1023 = 3 \\\\cdot 11 \\\\cdot 31$, so it is one of 1, 3, 11, 31, 33, 93, 341, 1023. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📐 The conjugation map The map $t \\\\mapsto t^{32}$ is a field automorphism of GF(1024). In coordinates it is $(a + bz) \\\\mapsto (a+b) + bz$. It fixes exactly the elements of GF(32), it swaps the two roots of $x^2 + x + 1$, and applying it twice returns the original element. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📐 Any single transcription error is detected If a valid codex32 string is transcribed with one wrong symbol, the result fails verification. The wallet that would have been silently wrong is loudly wrong instead. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📖 The bech32 alphabet The 32 characters, in field-value order: qpzry9x8gf2tvdw0s3jn54khce6mua7l The character at position $i$ in that string represents the field element $i$, counting from zero. From: codex32 Learn more:

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📐 Creation always produces a string that verifies Any string assembled by appending the checksum that the creation procedure produces passes verification. In the formalization this is Checksum.verify_create; the companion result encoded_seed_checksum_valid carries it through the encoder, proving that every serialized master seed has a valid checksum. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📖 Weight The weight of a polynomial is the number of its nonzero coefficients. A polynomial of weight at most 8 is one that differs from zero in at most eight places, however spread out. From: codex32 Learn more:

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📐 The long-only verifier is strictly more permissive There are strings that Checksum.verifyLong accepts and that Checksum.verify rejects. The BIP gives one explicitly. Proof: See the collapsible proof in the section itself, where it is presented in full. From: codex32 Learn more:

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📖 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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🏆 Magic Internet Math - Daily Top 10 🥇 Ape Mithrandir - 4,649 XP | Level 10 🥈 Anonymous - 3,632 XP | Level 9 🥉 Anonymous - 1,360 XP | Level 6 4. Pitufo - 211 XP | Level 2 5. Anonymous - 178 XP | Level 2 6. Lakesurfer - 10 XP | Level 1 7. Marc - 10 XP | Level 1 8. Gustavo - 10 XP | Level 1 9. Anonymous - 10 XP | Level 1 10. bassload@tutanota.com - 0 XP | Level 1 📚 Start learning:

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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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