Number Theory Lecture 8

The lecture NumTh8

Excerpts

Profinite groups: inverse limit of fintie groups.  Some common examples

  1. profinite completion \hat G := \varprojlim_i G/H_i, H_i all normal subgroups of finite index.
  2. pro-p completion \hat G_p := \varprojlim_i G/F_i, F_i all normal subgroups of index a power of p.

Another example.  For E/F an algebraic extension.  Look at all finite Galois extensions K_i/F \subset E/F.  These for a directed system under K_1 \le K_2 if K_1 \subset K_2.  Now we said earlier if F \subset K_1 \subset K_2 are all Galois extensions, then elements of Gal(K_2/F) restrict to Gal(K_1/F).  So we get an inverse system.  And infact

Gal(E/F) \cong \varprojlim_i Gal(K_i/F)

Also there is the fundamental theorem of Galois theory that says closed subgroups of a Galois group G = Gal(E/F) are in bijective correspondence with intermediate fields: E \supset K \mapsto Gal(E/K) \subset G.

The rest of the lecture gives an example of how this formalism occurs in previous constructions.  In particular another view of the contruction of the Frobenius element.

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