# Equality field extensions

How would you prove this: If E is an extension of K and a and b are elements of E\K, a^m belongs to K, b^n belongs to K, gcd (m, n) =1, then K(ab) =K(a,b) ?? I don't know which strategy to use to prove it.

## 1 Answer

We can assume that $$m$$ and $$n$$ are minimal such that $$a^m,b^n \in K$$ (if not replace $$m$$ and $$n$$ by the minimal such. They will still be coprime). Since $$m$$ and $$n$$ are coprime, there exists a $$k,l \in \mathbb{N}$$ such that $$mk = 1 + ln$$. Then $$(ab)^{mk} = (b^n)^l(a^m)^kb \in K(ab)$$ so since $$a^m,b^n \in K$$, this shows that $$b \in K(ab)$$. Similarly we can show that $$a \in K(ab)$$ (just consider $$(ab)^{nl}$$) and hence $$K(a,b) \subseteq K(ab)$$. Since $$K(ab) \subseteq K(a,b)$$, they must be equal.

You could also use degrees to prove this : if you can show that both $$[K(ab):K]=mn=[K(a,b):K]$$, then since $$K(ab) \subseteq K(a,b)$$, they must be equal.

• Ok, thank you very much for your answer! My previous attempts were to show what you stated below, but it is harder – Luis Gimeno Sotelo Dec 30 '18 at 12:33