Splitting of prime ideals in a number field

Suppose $𝐿 : 𝐾$ is an extension of number fields and $𝔭 ⊴ O 𝐾$ is a prime ideal of the ring of integers. Then by UFI, $𝔭 O 𝐿$ has a unique factorization into prime ideals $𝔭 𝑗 ⊴ O 𝐿$

𝔭 O 𝐿 = 𝑔 \prod 𝑗 = 1 𝔭 𝑒 𝑗 𝑗

where the multiplicities $𝑒 𝑗$ are called ramification indices. #m/def/num/alg Moreover,

if $𝔭 O 𝐿$ is a prime ideal, then $𝐿 : 𝐾$ is inert at $𝔭$ ;
if $𝑒 𝑗 > 1$ for some $𝑗$ , then $𝐿 : 𝐾$ is ramified at $𝔭$ ;
otherwise $𝐿 : 𝐾$ is unramified at $𝔭$ .

A fundamental result is Kummer's factorization theorem.

Properties

Let $𝐾 = ℚ (𝜗)$ be a number field. Then¹

If a minimal polynomial $𝑚 𝜗 (𝑥)$ is Eisenstein at $𝑝$ , then $𝑝$ is totally ramified in $O 𝐾$ .
If $𝑝$ does not divide the annoying index, then $𝑝$ ramifies in $O 𝐾$ iff $𝑝 ∣ Δ 𝐾 : ℚ$ .
Only finitely many primes ramify ramify in $𝐾$ .

Proof of 1.

From ^P2, we know that $𝑝$ does not divide $| O 𝐾 / ℤ [𝜗] |$ . By Kummer's factorization theorem, $𝑚 𝜗 (𝑥) \equiv 𝑝 𝑥 𝑛$ implies $⟨ 𝑝 ⟩ = ⟨ 𝑝, 𝜗 ⟩ 𝑛$ , proving ^P1.

#state/tidy | #lang/en | #SemBr

2022. Algebraic number theory course notes, §2.3.1 , pp. 41–43 ↩

Splitting of prime ideals in a number field

Properties

Footnotes