Abstract
A criterion for a prime to be a common index divisor of a dihedral field of prime degree is given. This criterion is used to determine the index of families of dihedral fields of degrees 5 and 7.
Highlights
Let L be an algebraic number field of degree n
The element α ∈ OL is called a generator of L if L = Q(α)
The index of L is i(L) = gcd ind α | α is a generator of L
Summary
Let L be an algebraic number field of degree n. Let OL denote the ring of integers of L. The element α ∈ OL is called a generator of L if L = Q(α). The index of α is the positive integer indα given by. A positive integer > 1 dividing i(L) is called a common index divisor of L. Βn−1} is an integral basis for L, L is said to be monogenic. A field possessing a common index divisor is nonmonogenic. Let f (x) be an irreducible polynomial in Z[x] of odd prime degree q and suppose that Gal( f (x)) Dq (the dihedral group of order 2q). Let M be the splitting field of f (x).
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