Text | One of the central objects in number theory is the absolute Galois group Gal(Q'/Q), the group of automorphisms of the field of algebraic numbers fixing the field of rational numbers. The current trend to study Gal(Q'/Q) is through its linear representations over finitely generated free Zp-modules, where Zp denotes the ring of p-adic integers. These representations are the so-called p-adic Galois representations, which are essentially continuous group homomorphisms Ï:Gal(Q'/Q)-->GL_n(Zp). Such representations naturally occur in geometry as actions of Gal(Q'/Q) on the p^n torsion points of algebraic varieties such as elliptic curves. Still though, Ralph Greenberg indicates that given a prime p and an integer n>3, it is hard to come up with Galois representations with an open image, that is, informally speaking, with a big image. He suggests that if a number field K has a particular type of extension M dependent on a fixed prime p, one can obtain p-adic representations of Gal(M/K) of certain degrees n. Such a field K is called p-rational. Our study is to understand Greenberg's way of constructing Galois representations with 'big images' and to study the questions surrounding p-rationality of number fields along the path he opens. This should lead to experimental and theoretical experiments to further explore p-rationality, as well as to an extension of the range of applicability of Greenberg's results to other groups than GL_n. |
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