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Additional resources for Algebraic number theory (Math 784)

Example text

N be the n roots of f (x), and let β1 = β, β2 , . . , βm be the roots of g(x). Thus, β1 , . . , βm are the conjugates of β. The numbers h(α1 ), h(α2 ), . . , h(αn ) are called the field conjugates of β in Q(α). This terminology is not as misleading as it may seem as the following theorem shows. Theorem 37. Given the notation above, m|n and h(α1 ), . . , h(αn ) is some arrangement n of n/m copies of β1 , . . , βm . In other words, if F (x) = j=1 (x − h(αj )), then F (x) = g(x)n/m . , if n = m), then Q(α) = Q(β).

Qs are irreducibles (and, hence, primes) in R. Let j ∈ {1, 2, . . , r}. Since pj is prime, pj |qi for some i ∈ {1, 2, . . , s}. Write 44 qi = βj pj where βj ∈ R. Since qi is irreducible and pj is a nonunit, we deduce that βj is a unit. Multiplying through by βj in (∗) and cancelling βj pj with qi , the number of prime factors on each side of (∗) decreases by one. Continuing in this manner, we get that r = s and (after rearranging) qj = βj pj for each j ∈ {1, 2, . . , r} with βj a unit in R.

Since p = λp−1 κ divides each of c0 , c1 , . . , ck−1 , the right-hand side above can be written in the form λk+1 κ for some κ ∈ Z[ζ] ⊆ R. Therefore, ck = λκ . Taking norms, we obtain = N (ck ) = N (λ)N (κ ) = pN (κ ). cp−1 k On the other hand, Theorem 39 implies that N (κ ) ∈ Z. Hence, p|cp−1 , and we deduce k that p|ck . This completes the proof that p|ck for all k ∈ {0, 1, . . , p − 2} by induction, and the theorem follows. 41 • The discriminant of Q(ζp ) where p is an odd prime. Theorem 47 implies that {1, ζp , .

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Algebraic number theory (Math 784) by Filaseta M.


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