Quadratic Fields
Prime Decomposition in Quadratic Fields
How rational primes split, remain inert or ramify in a quadratic field, decided entirely by the Kronecker symbol.
Engineering / MathematicsQuadratic Fields2 min readKV-MATH-0600
Decomposition in a quadratic field requires no polynomial factorisation. A single Kronecker symbol evaluation decides the case, which makes prime handling in quadratic fields essentially free.
The three cases
| Kronecker symbol of the discriminant at p | Case | Factorisation |
|---|---|---|
| 1 | Split | Two distinct primes, each of norm p |
| -1 | Inert | p remains prime, norm p squared |
| 0 | Ramified | A single prime squared, norm p |
Constructing the primes
Building the primes above p
- Evaluate the symbolUsing the reciprocity-based algorithm — see symbol computation.
- Split caseSolve the congruence for a square root of the discriminant modulo p, giving two roots and hence two primes.
- Ramified caseThe single prime is generated by p and the appropriate root.
- Inert caseThe prime is generated by p alone.
Ramified primes
A prime ramifies exactly when it divides the discriminant. Since the discriminant is squarefree up to a factor of four, only finitely many primes ramify and they are immediately identified by trial division.
Splitting behaviour
Consequences for factoring
The same criterion governs which primes can divide values of a quadratic form, which is exactly what determines the factor base in the quadratic sieve. The connection is not an analogy: the sieve is working in a quadratic order.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 5.1. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.