Field Arithmetic - Fields That Are Defined By Their Absolute Galois Groups

Fields That 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

This group is isomorphic to the absolute Galois group of an arbitrary finite field. Also the absolute Galois group of the field of formal Laurent series C((t)) over the complex numbers is isomorphic to that group.

To get another example, we bring below two non-isomorphic fields whose absolute Galois groups are free (that is free profinite group).

  • 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 its origins in Riemann's existence theorem. For a field of arbitrary characteristic it is due to David Harbater and Florian Pop, and was also proved later by Dan Haran and Moshe Jarden.)
  • The absolute Galois group Gal(Q) (where Q are the rational numbers) is compact, and hence equipped with a normalized Haar measure. For a Galois automorphism s (that is an element in Gal(Q)) let Ns be the maximal Galois extension of Q that s fixes. Then with probability 1 the absolute Galois group Gal(Ns) is free of countable rank. (This result is due to Moshe Jarden.)

In contrast to the above examples, if the fields in question are finitely generated over 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 Q and let a: Gal(K) → Gal(L) be an isomorphism. Then there exists a unique isomorphism of the algebraic closures, b: Kalg → Lalg, that induces a.

This generalizes an earlier work of Jürgen Neukirch and Koji Uchida on number fields.

Read more about this topic:  Field Arithmetic

Famous quotes containing the words fields, defined, absolute and/or groups:

    “It’s a melody full of the laughter of children out after the rain.”
    —Dorothy Fields (1904–1974)

    “The human race is a zone of living things that should be defined by tracing its confines.”
    —Italo Calvino (1923–1985)

    “All beauties contain, like all possible phenomena, something eternal and something transitory,—something absolute and something particular. Absolute and eternal beauty does not exist, or rather it is only an abstraction skimmed from the common surface of different sorts of beauty. The particular element of each beauty comes from the emotions, and as we each have our own particular emotions, so we have our beauty.”
    —Charles Baudelaire (1821–1867)

    “... until both employers’ and workers’ groups assume responsibility for chastising their own recalcitrant children, they can vainly bay the moon about “ignorant” and “unfair” public criticism. Moreover, their failure to impose voluntarily upon their own groups codes of decency and honor will result in more and more necessity for government control.”
    —Mary Barnett Gilson (1877–?)