Quadratic Field - Prime Factorization Into Ideals

Prime Factorization Into Ideals

Any prime number p gives rise to an ideal pO_K in the ring of integers O_K of a quadratic field K. In line with general theory of splitting of prime ideals in Galois extensions, this may be

p is inert

(p) is a prime ideal

The quotient ring is the finite field with p2 elements: O_K/pO_K = F_p2

p splits

(p) is a product of two distinct prime ideals of O_K.

The quotient ring is the product O_K/pO_K = F_p × F_p.

p is ramified

(p) is the square of a prime ideal of O_K.

The quotient ring contains non-zero nilpotent elements.

The third case happens if and only if p divides the discriminant D. The first and second cases occur when the Kronecker symbol (D/p) equals −1 and +1, respectively. For example, if p is an odd prime not dividing D, then p splits if and only if D is congruent to a square modulo p. The first two cases are in a certain sense equally likely to occur as p runs through the primes, see Chebotarev density theorem.

The law of quadratic reciprocity implies that the splitting behaviour of a prime p in a quadratic field depends only on p modulo D, where D is the field discriminant.