Ring of integers of a number field

Splitting of prime ideals in a number field

Suppose is an extension of number fields and is a prime ideal of the ring of integers. Then by UFI, has a unique factorization into prime ideals

where the multiplicities are called ramification indices. alg Moreover,

• if is a prime ideal, then is inert at ;
• if for some , then is ramified at ;
• otherwise is unramified at .

A fundamental result is Kummer’s factorization theorem.

Properties

Let be a number field. Then¹

1. If a minimal polynomial is Eisenstein at , then is totally ramified in .
2. If does not divide the annoying index, then ramifies in iff .
3. Only finitely many primes ramify ramify in .

Proof of 1.

From ^P2, we know that does not divide . By Kummer’s factorization theorem, implies , proving ^P1.

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Footnotes

1. 2022. Algebraic number theory course notes, §2.3.1 , pp. 41–43 ↩