A question on Groups and Galois Theory

Question

I'm studying Abstract Algebra and I need to construct a cyclic extension of order $2^5 3^4 5^{10}$. I have no idea of how to do that. Could someone help me?

Thank you!

Answer 1

Note that $2^53^45^{10} + 1$ is prime (at least according to WolframAlpha).

If we let $\zeta_n$ denote a primitive $n$th root of unity, then the cyclotomic field $\mathbb{Q}[\zeta_n]$, which is a splitting field of $f(x) = x^n - 1$, has a Galois group isomorphic to $\mathbb{Z}_n^\times$.

When $n$ is prime, then it is a theorem that $\mathbb{Z}_n^\times \cong \mathbb{Z}_{n-1}$.

Answer 2

You can look into finite fields. If $q$ is a prime power, then the extension $\Bbb F_q \subset \Bbb F_{q^N}$ is cyclic of order $N$ (it's generated by the $q$th power automorphism $x \mapsto x^q$).

So you can just pick $q=2$ and $N = 2^5 3^4 5^{10}$