Re: Can any one help?

In article <1119019888.322296.220510@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Babak" <babakarj@xxxxxxxxx> writes:
>Can any one help me with these two questions?
>
>1.
>Let K be a field, a is in K and p is a prime number, show that:
>x^p-a in K[x] over K is irreducible <=> in K has no root

=> is easy.
If it has no root, then adjoining one gives an extension of degree p, so
the degree of the splitting field of x^p-a over K must be divisible by p,
but if it was reducible then it wouldn't be.

>2.
>F is a finite field and Char(F)=p. If N:F->F , N(a)=a^p is a Frobenius
>automorphism, show that Aut(F) is a cyclic group and Aut(F) = <N>

Show that N has order n, where |F| = p^n.

Derek Holt.
.

Relevant Pages

  • Re: My problem with Fields
    ... As we have a splitting field, the extension is finite. ... Arturo Magidin ... Prev by Date: ...
    (sci.math)
  • Re: cyclotomic field is galois
    ... how would you prove then that the splitting field of a cyclotomic ... extension" and what it is you consider to be "derived equivalences." ... the splitting field of some polynomials over some given field is ... knowledge of the automorphisms of the extension. ...
    (sci.math)
  • Re: cyclotomic field is galois
    ... how would you prove then that the splitting field of a cyclotomic ... extension" and what it is you consider to be "derived equivalences." ... that the relevant automorphism group have the same order, when finite, ... the splitting field of some polynomials over some given field is ...
    (sci.math)
  • Re: cyclotomic field is galois
    ... how would you prove then that the splitting field of a cyclotomic ... That depends on what you consider to be the "definition" of "Galois ... extension" and what it is you consider to be "derived equivalences." ... a polynomials, so "is the splitting field of a ...
    (sci.math)
  • Re: normal extensions - splitting fields
    ... |isn't a galois extension and it ain't a separable extension. ... The fact that E is a splitting field for a polynomial f over F ... P with coefficients in F has a root r in a normal extension E, ...
    (sci.math)
Loading