Group and Field Theoretic Foundations

Every field k comes accompanied by a canonical Galois extension: the separable algebraic closure ¯k| k. Its Galois group G_k = G(k\k) is called the absolute Galois group of k. This extension has infinite degree in almost all cases, but it has the big advantage of consolidating within it all the various finite Galois extensions of k. For this reason one would like to put ¯k| k into the forefront of Galois theoretic considerations. But one is then faced with the problem that the main theorem of Galois theory does not hold true anymore in the usual sense. We explain this by the following Example. The absolute Galois group $${G_{{\rm{I}}{{\rm{F}}_p}}} = G({\rm{I}}{{\rm{\bar F}}_p}|{\rm{I}}{{\rm{F}}_p})$$ of the field IF_p of p elements contains the Frobenius automorphism ϕ which is defined by $${x^\varphi } = {x^p}\,{\rm{for}}\,{\rm{all}}\,x \in {\rm{I}}{{\rm{\bar F}}_p}$$.