Capstok — learn by doing

Why this matters

A field is a group's richer cousin: you get two compatible operations (addition and multiplication), each behaving well, plus a distributive law gluing them together. ZK proofs live almost entirely inside finite fields $\mathbb{F}_p$ (and their extensions $\mathbb{F}_{p^k}$ ) because every arithmetic statement we want to prove — 'I know $w$ such that $C(w) = 0$ ' — gets encoded as a system of polynomial equations over a field. You need addition to combine constraints, multiplication to compose them, and inverses so that division is unambiguous. The field axioms aren't ceremony; they're the contract that makes 'arithmetisation' work. Pick the wrong structure and SNARK soundness collapses.

Demo

A field $\mathbb{F}$ is a set with two operations $+, \cdot$ such that $(\mathbb{F}, +)$ is an abelian group with identity 0, $(\mathbb{F} \setminus \{0\}, \cdot)$ is an abelian group with identity 1, and multiplication distributes over addition. For prime , is a finite field of size .

(\mathbb{F}, +): \text{abelian group, identity } 0. \quad (\mathbb{F}^*, \cdot): \text{abelian group, identity } 1. \quad a(b+c) = ab + ac.\\ \mathbb{F}_p \text{ is a field iff } p \text{ is prime.}

Try it yourself

Run the snippet; confirm the inverse table is consistent: every $a \cdot a^{-1} \equiv 1 \pmod{13}$ . List the inverses of $2, 3, 4, 5, 6$ — these are the 'fractions' that ZK protocols compute as group exponents.

Prompt your AI

Use these three in order. Each builds on the one before.

1. Basics & terminology

In one paragraph, explain the difference between a *ring*, a *field*, and a *group* using $\mathbb{Z}, \mathbb{Q}, \mathbb{Z}/6\mathbb{Z}$ as running examples. Why is 'every nonzero element has a multiplicative inverse' the crucial extra property a field has?

2. Why it works (the mechanism)

Walk me through why $\mathbb{Z}/p\mathbb{Z}$ is a field exactly when $p$ is prime. Where in the proof does primality enter, and what concretely goes wrong in $\mathbb{Z}/6\mathbb{Z}$?

3. Advanced — application & what's next

ZK protocols often pick a specific 'SNARK-friendly' prime like the BN254 scalar field or BLS12-381's $r$. What properties of $p$ — beyond being prime — make a field SNARK-friendly? (Think about FFTs, two-adicity, low-degree extensions.)

References

p

p

// main.go
package main

import (
	"fmt"
)

// Working in F_p with p = 13
const p = 13

func add(a, b int) int { return (a + b) % p }
func mul(a, b int) int { return (a * b) % p }
func inv(a int) int {
	// Fermat: a^(p-2) is the inverse of a mod p when p is prime
	result := 1
	base := a % p
	exp := p - 2
	for exp > 0 {
		if exp%2 == 1 {
			result = (result * base) % p
		}
		base = (base * base) % p
		exp /= 2
	}
	return result
}

func main() {
	// Verify field axioms by sampling
	for a := 1; a < p; a++ {
		if mul(a, inv(a)) != 1 {
			panic(fmt.Sprintf("inverse of %d broken", a))
		}
	}
	// Distributivity spot-check
	a, b, c := 5, 7, 11
	if mul(a, add(b, c)) != add(mul(a, b), mul(a, c)) {
		panic("distributivity broken")
	}

	inverseTable := make(map[int]int)
	for a := 1; a < p; a++ {
		inverseTable[a] = inv(a)
	}
	fmt.Printf("Inverse table: %v\n", inverseTable)
	fmt.Println("Note: 12 mod 13 has inverse 12, since 12 == -1.")
}

Run: go run main.go

Fields and finite fields F_p