## Local class field theoryThis book provides a readable introduction to local class field theory, a theory of algebraic extensions. It covers abelian extensions in particular of so-called local fields, typical examples of which are the p-adic number fields. The book is almost self-contained and is accessible to any reader with a basic background in algebra and topological groups. |

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### Contents

Valuations | 3 |

Local Fields | 18 |

Infinite Extensions of Local Fields | 35 |

Copyright | |

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abelian group algebraic extension canonical Chapter class field theory commutative complete field contained Corollary of Proposition cyclic extension cyclic group exists a unique extension k'lk extension of degree Ff(X fields and let finite abelian extension finite extension finite field finite Galois extension fixed field follows from Lemma formal group Frobenius automorphism Furthermore Gal(kab/k Gal(kur/k Gal(L Gal(L/K implies induces an isomorphism integer jr-sequence Lemma Let f let f(X Let G Let k'lk maximal ideal mod deg n-sequence norm map norm residue map normalized valuation normed sequence o-module obtain p-adic p-field of characteristic pk(n pk(x polynomial prime element Proof Proposition 1.5 Proposition 2.12 prove ramification groups remark residue class residue field roots of unity satisfies separable extension subfield subgroup of kx Theorem 6.9 totally ramified extension Un+1 unique power series v-topology valuation ring Zp-module