MagicInternetMath
Magic Internet Math

https://mathacademy-cyan.vercel.app

Daily Insights from Magic Internet Math courses. Learn at https://mathacademy-cyan.vercel.app

Public Key

npub14m9zwklz4qfwveanne5qptrn0npgya3rxf6xxhpjpnwhzutdn2pqy6f2c4

NIP-05 Address

Profile Code

nprofile1qqs2aj38t032syhxv7eeu6qq43ehes5zwc3nyarrtseqeht3w9ke4qsppamhxue69uhkummnw3ezumt0d5qs6amnwvaz7tmwdaejumr0dsq3gamnwvaz7tmjv4kxz7fwv3sk6atn9e5k7pkgxtm

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nostr.mom

nos.lol

relay.damus.io

Author Public Key

npub14m9zwklz4qfwveanne5qptrn0npgya3rxf6xxhpjpnwhzutdn2pqy6f2c4

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Last Notes

2026-10-05 16:24:23 UTC

📐 The five constants are one generator scaled by a basis
Let $g$ be the generator, packed as $G_0 = \\\\texttt{0x19dc500ce73fde210}$. Then $G_i$ is $g$ with every coefficient multiplied by $2^i$ in GF(32), and for a departing symbol $t$ the XOR of the selected constants equals $t \\\\cdot g$.
Proof: See the collapsible proof in the section itself, where it is presented in full.
From: codex32
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2026-10-05 07:42:05 UTC

📖 Master seed format
Generic codex32, plus the requirement that the payload be one of the six application lengths: 26, 32, 39, 45, 52 or 103 symbols — corresponding to master seeds of 16, 20, 24, 28, 32 or 64 bytes.
From: codex32
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2026-10-05 01:18:01 UTC

📖 The de Bruijn criterion
A proof assistant satisfies the de Bruijn criterion if all proofs it accepts are ultimately certified by a small, fixed kernel whose only job is to check terms. All the convenient machinery — automation, tactics, decision procedures — must produce a term the kernel independently accepts, and is therefore not trusted itself.
From: codex32
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2026-10-04 22:32:57 UTC

📖 Polynomial over GF(32)
A polynomial over GF(32) is a finite expression $f(X) = a_0 + a_1 X + a_2 X^2 + \\\\cdots + a_d X^d$ in which every coefficient $a_i$ is an element of GF(32). If $a_d \\\\ne 0$, the degree of $f$ is $d$. The zero polynomial has no degree.
From: codex32
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2026-10-04 18:48:30 UTC

📐 Eight-symbol detection
Any two equal-length valid codex32 strings within a checksum period that differ in at most eight symbols are in fact the same string.
Proof: See the collapsible proof in the section itself, where it is presented in full.
From: codex32
Learn more: https://mathacademy-cyan.vercel.app/#/section/33
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2026-10-04 14:59:14 UTC

📐 ℤp is a field when p is prime
For $p$ prime, $\\\\mathbb{Z}_p$ is a field with $p$ elements.
Proof: See the collapsible proof in the section itself, where it is presented in full.
From: codex32
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2026-10-04 08:49:11 UTC

📐 A field has no zero divisors
If $ab = 0$ in a field and $a \\\\ne 0$, then $b = 0$.
Proof: See the collapsible proof in the section itself, where it is presented in full.
From: codex32
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2026-10-04 02:00:01 UTC

📖 Derivation
Computing the share at an index not currently held, by interpolating the held shares at that index — column by column, exactly as in Part 3.
From: codex32
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2026-10-03 21:47:21 UTC

📐 A one-symbol one-time pad over GF(32) has perfect secrecy
With a uniformly random key used once, every ciphertext is equally likely under every message, so observing the ciphertext leaves the attacker\
Proof: See the collapsible proof in the section itself, where it is presented in full.
From: codex32
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2026-10-03 18:17:55 UTC

📐 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
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2026-10-03 13:56:27 UTC

📐 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
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2026-10-03 08:34:20 UTC

📐 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
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2026-10-03 02:20:02 UTC

📖 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
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2026-10-02 23:16:37 UTC

📐 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
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2026-10-02 19:32:35 UTC

📖 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
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2026-10-02 14:09:14 UTC

📐 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
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2026-10-02 07:32:15 UTC

📖 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
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2026-10-02 01:05:29 UTC

📖 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
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2026-10-01 21:24:14 UTC

📖 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
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2026-10-01 18:07:41 UTC

2026-10-01 16:42:27 UTC

📐 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
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2026-10-01 09:26:31 UTC

📖 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
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2026-10-01 02:27:10 UTC

📖 Zero divisor
A nonzero element $a$ is a zero divisor if there is some nonzero $b$ with $a \\\\times b = 0$.
From: codex32
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2026-09-10 05:19:34 UTC

📖 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
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2026-09-10 00:37:11 UTC

📖 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
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2026-09-09 21:58:03 UTC

📐 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
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2026-09-09 18:51:37 UTC

📐 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
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2026-09-09 14:48:50 UTC

📐 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
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2026-09-09 10:08:14 UTC

📐 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
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2026-09-09 05:18:26 UTC

📖 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
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2026-09-09 00:38:53 UTC

📐 Cayley-Hamilton Theorem
If $T \\in \\mathcal{L}(V)$ and $q$ is the characteristic polynomial of $T$, then $q(T) = 0$.
From: linalg-axler
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2026-09-08 22:02:22 UTC

📖 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
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2026-09-08 18:58:46 UTC

📖 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
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2026-09-08 14:47:09 UTC

📐 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
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2026-09-08 10:06:15 UTC

📖 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
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2026-09-08 05:16:48 UTC

📐 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
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2026-09-08 00:41:09 UTC

📖 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
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2026-09-07 21:47:27 UTC

🔗 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
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2026-09-07 18:25:31 UTC

📖 Field
A field $F$ is a set which, under addition, forms an additive abelian group, and for which the nonzero elements form a multiplicative group. The two operations are connected by the distributive law $a(b+c) = ab + ac$ and $(b+c)a = ba + ca$.
From: gal-artin
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2026-09-07 13:01:50 UTC

📖 Cycle Notation
The mapping $T(i) = j, T(j) = k, \\ldots, T(m) = i$ is written $(i\\,j\\,\\ldots\\,m)$ and called a $k$-cycle. A 2-cycle $(i\\,j)$ is a transposition.
From: gal-artin
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2026-09-07 06:38:27 UTC

🎮 Interactive: Pell Equation Solver
Find integer solutions to x^2 - Dy^2 = 1. Watch continued fractions reveal the fundamental solution to these ancient Diophantine equations.
From: Disquisitiones Arithmeticae
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2026-09-07 01:18:51 UTC

📐 Galois
Let $f(x) = 0$ be an equation with distinct roots whose Galois group over $K$ is $G$. Then $f(x) = 0$ can be solved by radicals if and only if $G$ is solvable -- that is, has a composition series $G \\supset G_1 \\supset G_2 \\supset \\cdots \\supset G_\\nu = \\{e\\}$ in which each $G_i$ is a normal subgroup of prime index in its predecessor.
Proof: Necessity: If solvable by radicals, the tower of field extensions reduces the Galois group at each step to a normal subgroup of prime index (by the proposition of \u00a744). Taking only steps where the group decreases gives the composition series. Sufficiency: If $G$ is solvable, the proposition ...
From: gal-edwards
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2026-09-06 23:13:18 UTC

📐 Structure of Roots of Unity
To find primitive $n$th roots of unity for an arbitrary positive integer $n$, it suffices to be able to find primitive $p$th roots of unity for each prime $p$. This is because any $n$ can be written as a product of prime powers, and the $p^{n-1}$st root of a primitive $p$th root of unity is a primitive $p^n$th root.
Proof: When $n = jk$ where $j$ and $k$ are relatively prime, the product of a primitive $j$th root and a primitive $k$th root is a primitive $n$th root. More generally, if $n = j \\cdot k \\cdots m$ with pairwise coprime factors, the primitive $n$th roots of unity are products of primitive roots of the ...
From: gal-edwards
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2026-09-06 21:44:10 UTC

📖 Elementary Symmetric Functions
For the polynomial $(t - x_1)(t - x_2) \\cdots (t - x_n) = t^n + a_1 t^{n-1} + \\cdots + a_n$, the coefficients $a_i = (-1)^i$ times the sum of all products of $i$ different variables are the elementary symmetric functions.
From: gal-artin
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2026-09-06 19:31:26 UTC

📖 Solvable by Radicals
A polynomial $f(x) \\in F[x]$ is **solvable by radicals** if there is a tower $F = F_0 \\subset F_1 \\subset \\cdots \\subset F_n$ with $F_n$ containing all roots of $f$, where each $F_{i+1} = F_i(\\alpha_i)$ with $\\alpha_i^{n_i} \\in F_i$ for some $n_i$.
From: gal-morandi
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2026-09-06 17:21:35 UTC

📖 Dependence of Characters
Characters $\\sigma_1, \\ldots, \\sigma_n$ are dependent if there exist $a_1, \\ldots, a_n \\in F$, not all zero, such that $a_1 \\sigma_1(x) + \\cdots + a_n \\sigma_n(x) = 0$ for each $x \\in G$. Otherwise they are independent.
From: gal-artin
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2026-09-06 14:24:50 UTC

📖 Transcendental Extension
An element $a$ is transcendental over a field $K$ if it satisfies no polynomial equation with coefficients in $K$. The extension $K(a)$ is then a transcendental extension. Polynomials with coefficients in $K(a)$ can be regarded as polynomials in two variables ($a$ and $x$) with coefficients in $K$.
From: gal-edwards
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2026-09-06 11:21:48 UTC

🎮 Interactive: Isometry Group Demo
Discover the isometries of 3D space: translations, rotations, reflections, and glide reflections. Every distance-preserving map is one of these!
From: Four Pillars of Geometry
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2026-09-06 06:31:12 UTC

📐 Kronecker
Let $f(x)$ be a polynomial of degree $n > 0$ with integer coefficients. If $f(x)$ has a nontrivial factorization over $\\mathbb{Z}$, then one of the factors has degree $\\leq n/2$. A factor of degree $m$ is determined by its values at $m + 1$ points and must divide $f(j)$ at each integer $j$. This gives a finite search using Lagrange interpolation.
From: gal-edwards
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2026-09-06 01:20:51 UTC

🔗 Hensel
Let $K$ be a complete valued field with valuation ring $\\mathcal{O}$ and residue field $k$. If $f(x) \\in \\mathcal{O}[x]$ and $\\bar{f}(x) = \\bar{g}(x)\\bar{h}(x)$ in $k[x]$ with $\\gcd(\\bar{g}, \\bar{h}) = 1$, then there exist $g(x), h(x) \\in \\mathcal{O}[x]$ lifting $\\bar{g}, \\bar{h}$ with $f = gh$.
From: gal-jacobson
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2026-09-05 23:22:10 UTC

📐 Division Algorithm for Polynomials
If $f, g \\in F[X]$ with $g \\neq 0$, there exist unique $q, r \\in F[X]$ with $f = qg + r$ and $\\deg(r) < \\deg(g)$.
From: gal-weintraub
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2026-09-05 21:43:07 UTC

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