Degree of extension over finite fields with two elements having same degree irreducible

Question

I get that Q(√2,√3) has degree 4 over Q. Now consider finite field F and a,b lying in an extension of F having same degree irreducible polynomial over F, let it be 'd'. Is the extension F(a,b), of degree d or d^2`? Like for example for d=2, whether the degree of extension is 2 or 4?

Answer 1

A finite field $F$ of size $q$ has only one extension of any given degree $d$, because there is only one field $H$ of size $q^d$, and there is only one copy of $F$ inside $H$, consisting of roots of the polynomial $x^q-x$.

Thus, if $F(a)$ and $F(b)$ both have degree $d$, we have in fact $F(a)=F(b)=F(a,b)$, hence $F(a,b)$ also has degree $d$.

By the way, I don’t know where you got $2d$ from in the first place. If $F$ is an arbitrary field, the degree of $F(a,b)$ is a multiple of $d$ between $d$ and $d^2$. The exact value depends on the situation, but there is no particular reason for it to be $2d$. More generally, if $d=[F(a):F]$ and $e=[F(b):F]$, then $[F(a,b):F]$ is a multiple of $\operatorname{lcm}(d,e)$ between $\operatorname{lcm}(d,e)$ and $de$.