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:
“The worlds great day is growing late,
Yet strange these fields that we have planted
So long with crops of love and hate.”
—Edwin Muir (18871959)
“Under the rules of a society that cannot distinguish between profit and profiteering, between money defined as necessity and money defined as luxury, murder is occasionally obligatory and always permissible.”
—Lewis H. Lapham (b. 1935)
“The teacher must derive not only the capacity, but the desire, to observe natural phenomena. In our system, she must become a passive, much more than an active, influence, and her passivity shall be composed of anxious scientific curiosity and of absolute respect for the phenomenon which she wishes to observe. The teacher must understand and feel her position of observer: the activity must lie in the phenomenon.”
—Maria Montessori (18701952)
“Under weak government, in a wide, thinly populated country, in the struggle against the raw natural environment and with the free play of economic forces, unified social groups become the transmitters of culture.”
—Johan Huizinga (18721945)