Abstract
AbstractWe study the fields of values of the irreducible characters of a finite group of degree not divisible by a primep. In the case where$p=2$, we fully characterise these fields. In order to accomplish this, we generalise the main result of [ILNT] to higher irrationalities. We do the same for odd primes, except that in this case the analogous results hold modulo a simple-to-state conjecture on the character values of quasi-simple groups.
Highlights
It is of great interest to study the values of the complex irreducible characters of a finite group
If (1) is odd and Q( ) = Q( ) for a square-free integer ≠ −1, ≡ 1. This is a consequence of the recent main result of [ILNT], whose proof uses the classification of finite simple groups: if (1) is odd, either Q( ) is contained in the cyclotomic field Q for some odd integer or ∈ Q( )
Before going into the main results of this paper, we prove that given any abelian extension /Q, where ⊆ C, there exist a finite group and ∈ Irr( ) such that = Q( )
Summary
It is of great interest to study the values of the complex irreducible characters of a finite group. If is a character (not necessarily irreducible) of a finite group , the field of values Q( ) of is the smallest field containing ( ) for all ∈. If (1) is odd and Q( ) = Q( ) for a square-free integer ≠ −1, ≡ 1 (mod 4) This is a consequence of the recent main result of [ILNT], whose proof uses the classification of finite simple groups: if (1) is odd, either Q( ) is contained in the cyclotomic field Q for some odd integer or ∈ Q( ). F = Q( )/Q | ∈ Irr ′ ( ), a finite group , where Irr ′ ( ) is the set of complex irreducible characters of of degree not divisible by ?
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