Let $K$ be a field of positive characterstic. I need to show that there exists $a$ and $b$ of the same degree over $K$ but $K(a)$ and $K(b)$ are not isomorphic.
I thought of an example where they are not $K$-isomorphic. Take $K=F_p(X)$ the function field in one variable over $F_p$, $a$ to be a root of a separable irreducible polynomial and $b$ a root of a non-separable one of the same degree.
How can I show that examples exist for which $K(a)$ and $K(b)$ are not isomorphic?
Best Answer
This is not true as stated. Indeed, if $K$ is a finite field, then for any positive integer $n$ there is a unique extension of $K$ of degree $n$, up to isomorphism (namely, the splitting field of $x^m-x$ where $m=|K|^n$). Or if $K$ is algebraically closed, then it has no nontrivial finite extensions at all.
For a simple example where you get to choose the field $K$, take $K=\mathbb{F}_p(X)$, let $K(a)=\mathbb{F}_p(\sqrt{X})$ and let $K(b)=\mathbb{F}_{p^2}(X)$ (so $a=\sqrt{X}$ and $b$ is some element of $\mathbb{F}_{p^2}\setminus\mathbb{F}_p$). These both have degree $2$ over $K$, but they are not isomorphic since $\mathbb{F}_{p^2}$ does not embed in $K(a)$ (which is isomorphic to $K$ by sending $\sqrt{X}$ to $X$).