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:
“Most books belong to the house and street only, and in the fields their leaves feel very thin. They are bare and obvious, and have no halo nor haze about them. Nature lies far and fair behind them all. But this, as it proceeds from, so it addresses, what is deepest and most abiding in man. It belongs to the noontide of the day, the midsummer of the year, and after the snows have melted, and the waters evaporated in the spring, still its truth speaks freshly to our experience.”
—Henry David Thoreau (1817–1862)
“Social history might be defined negatively as the history of a people with the politics left out.”
—G.M. (George Macaulay)
“I tell you, hopeless grief is passionless;
That only men incredulous of despair,
Half-taught in anguish, through the midnight air
Beat upward to God’s throne in loud access
Of shrieking and reproach. Full desertness,
In souls as countries, lieth silent-bare
Under the blanching, vertical eye-glare
Of the absolute Heavens.”
—Elizabeth Barrett Browning (1806–1861)
“screenwriter
Policemen so cherish their status as keepers of the peace and protectors of the public that they have occasionally been known to beat to death those citizens or groups who question that status.”
—David Mamet (b. 1947)