High Quality Content by WIKIPEDIA articles! 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. A pseudo algebraically closed field (in short PAC) K is a field satisfying the following geometric feature. Each absolutely irreducible algebraic variety V defined over K has a K-rational point. Over PAC fields there is a firm link between arithmetic properties of the field and group theoretic properties of its absolute Galois group. A nice theorem in this spirit connects Hilbertian fields with -free fields (K is -free if any embedding problem for K is properly solvable).

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