# Number Theory Lecture 8

The lecture NumTh8

### Excerpts

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

- profinite completion , all normal subgroups of finite index.
- pro-p completion , all normal subgroups of index a power of .

Another example. For an algebraic extension. Look at all finite Galois extensions . These for a directed system under if . Now we said earlier if are all Galois extensions, then elements of restrict to . So we get an inverse system. And infact

Also there is the fundamental theorem of Galois theory that says closed subgroups of a Galois group are in bijective correspondence with intermediate fields: .

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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You’re currently reading “Number Theory Lecture 8,” an entry on Math Meandering

- Published:
- October 12, 2009 / 6:35 pm

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- Number Theory (course)

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