Extensions with the same degree are $K-$isomorphic?

Question

Suppose we have two simple algebraic extensions $K(a)$, $K(b)$ over a field $K$

(I am interested in case $\mathrm{char}(K)=0$ but I guess there's no difference)

If they have the same (finite) degree over $K$, then they are always $K-$isomorphic?

Answer 1

If $[K(a):K]=[K(b):K]$, then $\dim_K K(a)=\dim_K K(b)<\infty$ by definition and hence they are isomorphic as $K$-vector spaces by elementary linear algebra. That is, there is an invertible $K$-linear map $K(a)\to K(b)$. If we want this map to be a $K$-isomorphism (i.e. fixing $K$) we can write down explict bases for $K(a)$ and $K(b)$

$$\mathfrak B_1=\{1,a,\dots,a^{n-1}\},\quad\mathfrak B_2=\{1,b,\dots,b^{n-1}\}$$

where $[K(a):K]=n=[K(b):K]$ and force $1\mapsto1$. By linearity this ensures that $K$ is fixed. This also works in the absense of a power basis as we can freely insert $1$ into any given basis.

It is, indeed, not necessary that $\operatorname{char} K=0$ for this to work.