Field arithmetic

In mathematics, Field Arithmetic is a subject that studies the interrelations between arithmetic properties of a field and its absolute Galois group. It is an interdisciplinary subject as it uses tools from algebraic number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and of profinite groups.

Fields with Finite Absolute Galois Groups

Let $K$ be a field and let $G=Gal(K)$ be its absolute Galois group. If $K$ is algebraically closed, then $G=1$ . If $K=\mathbb {R}$ , then $G=Gal(\mathbb {C} /\mathbb {R} )=\mathbb {Z} /2\mathbb {Z}$ . A theorem of Artin-Schreier asserts that (essentially) these are all the possibilities for finite absolute Galois groups.

Artin-Schreier Theorem. Let $K$ be a field whose absolute Galois group $G$ is finite. Then either $K$ is algebraically closed and $G$ is trivial or $K$ is real closed and $G=\mathbb {Z} /2\mathbb {Z}$ .

Fields Which are Defined by Their Absolute Galois Groups

Some profinite groups occur as the absolute Galois group of non-isomorphic fields. A first example for this is ${\hat {\mathbb {Z} }}=\lim _{\leftarrow }\mathbb {Z} /n\mathbb {Z}$ which is isomorphic to the absolute Galois group of an arbitrary finite field.

To get another example, we bring below two non-isomorphic fields whose absolute Galois groups are isomorphic to the free profinite group of countable rank which we denote by ${\hat {F}}_{\omega }$ :

Let $C$ be an algebraically closed field and $x$ a variable. Then $Gal(C(x))$ is free of rank equal to the cardinality of $C$ . (This result is due to Adrien Douady for 0 characteristic and has it origins in Riemann's Existence Theorem. For a field of arbitrary characteristic it is due to David Harbater and Florian Pop.)

The absolute Galois group $Gal(\mathbb {Q} )$ is compact, and hence equipped with a normalized Haar measure. For $\sigma \in Gal(\mathbb {Q} )$ let $N_{\sigma }$ be the maximal Galois extension of $\mathbb {Q}$ that $\sigma$ fixes. Then for almost all $\sigma \in Gal(\mathbb {Q} )$ we have $Gal(N_{\sigma })\cong {\hat {F}}_{\omega }$ . (This result is due to Moshe Jarden.)

In contrast to the above examples, if the fields in question are finitely generated over $\mathbb {Q}$ , Florian Pop proves that an isomorphism of the absolute Galois groups yields an isomorphism of the fields:

Theorem. Let $K,L$ be finitely generated fields over $\mathbb {Q}$ and let $\alpha \colon Gal(K)\to Gal(L)$ an isomorphism. Then there exists a unique isomorphism of the algebraic closures, $\phi \colon {\tilde {K}}\to {\tilde {L}}$ , that induces $\alpha$ .