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?
Asked
Active
Viewed 77 times
0
-
1Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – José Carlos Santos May 05 '20 at 06:29
1 Answers
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 $de$.
Emil Jeřábek
- 1,310