Adding an imaginary unit to a finite field
6 days ago
- #finite fields
- #Ethereum
- #elliptic curves
- Integers modulo a prime p form a finite field.
- Finite fields must have an order that is a power of a prime (q = p^n).
- For n > 1, field elements can be represented as polynomials of degree n-1 with coefficients in integers mod p.
- Addition in the field is coefficient-wise modulo p, while multiplication involves polynomial multiplication followed by division by an irreducible polynomial.
- For n=2, some fields can be constructed by adjoining an imaginary unit i, similar to complex numbers.
- A field of order p^2 can be formed by adjoining i if and only if p ≡ 3 mod 4.
- Example: Ethereum uses a field Fp[i] where p ≡ 3 mod 4 for elliptic curve operations.
- A specific point on the alt_bn128 curve in Ethereum is verified using Python code for field operations.