The Splitting Field of X3 − 2 over Q
نویسنده
چکیده
In Ok, the rational prime 2 has the principal prime factorization (2) = ( 3 √ 2)3, so k has class number 1: the ring Ok has unique factorization. Next we show Ok = Z[ 3 √ 2]. Let α = a+b 3 √ 2+c 3 √ 4 be an algebraic integer, with a, b, c all rational. Computing Trk/Q of α, α 3 √ 2, and α 3 √ 4 we see 3a, 6b, 6c ∈ Z. So the denominators of a, b, and c involve at most 2 and 3. To show 2 and 3 do not appear in the denominator, we consider the situation p-adically for p = 2 and p = 3.
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