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Dedekind factorization theorem

If

O_{K} = Z [θ]

and the minimal polynomial of

θ

factors modulo a rational prime

p

as

\overline{f} = \prod_{i} \overline{g}_{i}^{e_{i}},

then

(p) = \prod_{i} (p, g_{i} (θ))^{e_{i}},

and the prime ideal

(p, g_{i} (θ))

has norm

p^{d e g g_{i}}

.

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