Kronecker-Weber Theorem and Finite Fields

Question

Today it occurred to me that every algebraic extension of $\mathbb F_q$ is cyclotomic, as $\mathbb F_{q^n}$ can be gotten by adjoining a $(q^n-1)^{st}$ root of unity. Also, every algebraic extension of finite fields is abelian. Thus, a version of the Kronecker-Weber theorem holds, but it does so trivially. Is there anything important that can be seen from this observation, or is it just silly?

Answer 1

This is an important fact when studying extensions of local fields, say the field of $p$-adic numbers $\mathbf Q_p$.

Finite unramified extensions of a local field correspond to finite extensions of the residue field, and the respective Galois groups are isomorphic. There is a unique unramified extension of $\mathbf Q_p$ of degree $f$. If we write $q=p^f$, then we can write $\mathbf Q_q$ for the splitting field of $x^{q}-x$ over $\mathbf Q_p$; it is, up to (non-unique) isomorphism, the unique unramified extension of degree $f$ of $\mathbf Q_p$. We have $\text{Gal}(\mathbf Q_q/\mathbf Q_p) = \text{Gal}(\mathbf F_q/\mathbf F_p) = \mathbf Z/f\mathbf Z$.

Compare to the global side, where a number field only has finitely many unramified abelian extensions.

Thus a local field always has a distinguished collection of unramified abelian extensions, which are conceptually very simple. A useful property is also that their Galois groups are pointed groups, since they are equipped with a canonical generator (the Frobenius). These play a special role throughout all of algebraic number theory.

(Question for you: can you describe the action of Frobenius on $\mathbf Q_q$? We are in characteristic $0$, so it cannot be true anymore that $(x+y)^p = x^p + y^p$, though it is true mod $p$...)