Re: roots of polynomials on a field

From: Timothy Murphy <tim@xxxxxxxxxxxxxxxxxxxxxx>
Date: Sun, 19 Jun 2005 12:39:21 +0100

Li Yi wrote:

> We know that a polynomial of degree n has at most n roots in a field.
>>>From the finite field theory, we know that the splitting field of
> x^(p^n)-x on F(char F = p) has the order of p^n. The proof says that
> x^(p^n)-x has p^n distinct roots on F and then it says that those roots
> form a field. The conclusion follows.
>
> My question is, why does f(x) = x^(p^n)-x has p^n roots? I know that
> its roots are distinct since f and f' are coprime.

Because it is of degree p^n, and a polynomial p(x) of degree N over k
has N roots
(in the algebraic closure of k, or in the splitting field of p(x)).

--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
.

References:
- roots of polynomials on a field
  - From: Li Yi

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