What is(are) the Galois Extension(s) of $\mathbb{Q}$ whose Galois group is cyclic group of prime order? The fundamental theorem of Galois theory says that the degree of the extension is same as the order of the Galois group.Can we find an explicit polynomial of degree which is an arbitrary prime number?
Galois Extension of Q with Cyclic Group of Prime Order
galois-theory
Best Answer
To construct an Galois extension of $\mathbb{Q}$ of order $p$, take an integer $N$ such that $(\mathbb{Z}/N\mathbb{Z})^*$ maps surjectively onto $\mathbb{Z}/p\mathbb{Z}$, then the field $\mathbb{Q}(\zeta_N)$ contains a subfield $K$ such that $[K:\mathbb{Q}]=p$ and $K$ is Galois over $\mathbb{Q}$ (Here $\zeta_N$ is a primitive $N$-th root of unity). For example, to find a galois extension of $\mathbb{Q}$ of degree 5, you may look at the subfield of $\mathbb{Q}(\zeta_{11})$ that's fixed by the complex conjugation. Kronecker-weber Theorem states that every finite abelian extensions of $\mathbb{Q}$ is a subfield of cyclotomic fields, so every Galois extension of $\mathbb{Q}$ of prime order arises this way.
Continue the example with prime $p=5$, to find all of such extensions, we will look at the sequence $5n+1$. By Dirichlet's theorem on primes in arithmetic progressions, there are infinitely many primes $q$ in this arithmetic progression, each $\mathbb{Q}(\zeta_q)$ contains a subfield which is a degree 5 extension of $\mathbb{Q}$. All these subfields are distinct because they ramifies at different primes. Last, note that that there is also a subfield of $\mathbb{Q}(\zeta_{25})$ which is of degree 5 over $\mathbb{Q}$.
Edit: I was wrong about my complete list statement. For example, $\mathbb{Q}(\zeta_{341})$ (composite of $\mathbb{Q}(\zeta_{11})$ and $\mathbb{Q}(\zeta_{31})$) has 6 subfields which are degree 5 abelian extensions of $\mathbb{Q}$. Apologies.