0

Consider the following problem: from algebra by Thomas Hungerford Section 5.

If char $K = p \neq $0, let $K_p = ${ $u^p - u | u\in K$ } then show that a cyclic extension field F of K of degree p exists if and only if $K \neq K_p$ .

Let K $\neq K_p$ , but I couldn't make any progress on this side.

Now, for the opposite, it is given that Cyclic extension exists and let K =$K_p$ , so $u^p - u=u$ and I obtained a contradiction taht the polynomial is neither irreducible nor separable.

But I need help for the converse part.

Ty!

1 Answers1

0

Let $K\ne K_p$ and $a\in K\setminus K_p$, i.e. the polynomial $f=x^p-x-a$ has no roots in $K$. By this answer $f$ is irreducible. Let $L=K(u)$ where $u$ is a zero of $f$ in some extension field. A straightforward computation shows that $u+r$ for $r\in\Bbb F_p$ (the prime field) are also zeros of $f$ (or just see the linked answer again). It follows that $L/K$ is Galois and that the Galois group is generated by $u\mapsto u+1$, i.e. it is cyclic.

leoli1
  • 6,979
  • 1
  • 12
  • 36
  • Can you please tell why K is a bigger set than $K_p$? –  Oct 16 '21 at 16:47
  • 1 Can you please reply to my comment above? –  Jan 08 '22 at 18:32
  • Can you please reply to my above comment if you have some time to spare? Thanks! –  Aug 25 '23 at 20:27
  • It seems to me that the original question is asking: If $K\ne K_p$, then $K$ has a cyclic extension of degree $p$. So I am assuming that $K\ne K_p$. – leoli1 Aug 27 '23 at 01:06