Splitting of prime ideals in a number field

Kummer’s factorization theorem

Let $𝐾 = ℚ (𝜗)$ be a number field where $𝜗$ is an algebraic integer, and suppose $𝑝$ is a prime number not dividing the annoying index $| O 𝐾 / ℤ [𝜗] |$ . Let $𝑚 𝜗 (𝑥) \in ℤ [𝑥]$ be the minimal polynomial of $𝜗$ , and write

𝑚 𝜗 (𝑥) \equiv 𝑝 𝑔 \prod 𝑖 = 1 𝑓 𝑖 (𝑥) 𝑒 𝑖 .

for $𝑓 𝑖 (𝑥) \in ℤ [𝑥]$ ^irreducible mod $𝑝$ . Then

𝑝 O 𝐾 = 𝑔 \prod 𝑖 = 1 𝔭 𝑒 𝑖 𝑖

where $𝔭 𝑖 = ⟨ 𝑝, 𝑓 𝑖 (𝜗) ⟩$ are distinct prime ideals of norm $N (𝔭 𝑖) = 𝑝 d e g 𝑓 𝑖$ . alg We also have

𝑔 \sum 𝑖 = 1 𝑒 𝑖 d e g 𝑓 𝑖 = 𝑛

Proof

proof

Corollaries

See Splitting of prime ideals in a number field.

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Kummer’s factorization theorem

Corollaries

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