Splitting and ramification of primes

For a degree-n number field K and rational prime p, write $p\mathcal O_K=\prod_{i=1}^g\mathfrak P_i^{e_i}$ with distinct prime ideals. The residue field $\mathcal O_K/\mathfrak P_i$ has degree $f_i$ over $\mathbb F_p$ and cardinality $p^{f_i}$. The fundamental degree identity is $\sum_i e_if_i=n$. In a non-Galois number field, the e-values and f-values need not be identical across the primes; replacing the sum by one product without that hypothesis can give a false count.

Ramification means at least one exponent exceeds one. Complete splitting means every exponent and residue degree is one, while inertness means the rational prime ideal itself stays prime with residue degree n. These definitions concern the full integer ring. A quotient of order p squared need not be the field with p squared elements: ramification can instead produce nilpotents in the quotient by p. The separate prime-ideal quotients are fields, whereas the entire quotient by the rational prime can be a product or a nonreduced ring.

Let $\mathcal O_K$ contain the power order $\mathbb Z[\theta]$ for an integral primitive element θ, with monic minimal polynomial f. If p does not divide the order index, factor its reduction into distinct monic irreducibles $\overline f=\prod_i g_i^{e_i}$. Dedekind–Kummer identifies the primes as $(p,g_i(\theta))$, where an integer polynomial lift of each factor is used, and gives residue degree equal to the factor degree. Repeated factors then describe ramification under this index hypothesis.

If p divides the index, neither factor multiplicities nor the resulting ideal list can be trusted without an appropriate refinement. Replacing the generator, computing a maximal-order basis, or using a local method may resolve the issue. Squarefree reduction away from the index gives an unramified prime, but a repeated factor at an index prime can be spurious. The field discriminant supplies an independent ramification check, and the sum of e times f supplies a consistency check after a legitimate decomposition has been obtained.

For a quadratic field at an unramified odd prime, a split defining polynomial gives two degree-one residue fields, while an irreducible polynomial gives one degree-two residue field. When the maximal-order polynomial has a repeated root under the valid factorization criterion, one gets one prime with ramification index two. The norm identity remains the same in each case, but the arithmetic is different. Local completions will later reflect these alternatives as separate factors or as an extension with different ramification and residue behavior.

A quadratic congruence test must use the right generator and treat exceptional primes separately. It is also useful to record which root defines which residue map: a generator may reduce to different elements in the two split residue fields. Those maps are not merely names for two copies of the same quotient. Galois automorphisms can exchange the primes and their residue values, and the later Frobenius construction depends on this prime-by-prime viewpoint rather than on an undifferentiated list of polynomial roots.