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In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a 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'' = R is the real numbers, then : Here C is the field of complex numbers and Z is the ring of integer numbers. A theorem of Artin and 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 separably closed and ''G'' is trivial or ''K'' is real closed and ''G'' = Z/2Z. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Field arithmetic」の詳細全文を読む スポンサード リンク
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