This e-book reviews Hopf algebras over valuation earrings of neighborhood fields and their program to the speculation of wildly ramified extensions of neighborhood fields. the consequences, no longer formerly released in e-book shape, exhibit that Hopf algebras play a usual function in neighborhood Galois module idea. integrated during this paintings are expositions of brief specific sequences of Hopf algebras; Hopf Galois buildings on separable box extensions; a generalization of Noether's theorem at the Galois module constitution of tamely ramified extensions of neighborhood fields to wild extensions acted on through Hopf algebras; connections among tameness and being Galois for algebras acted on by way of a Hopf algebra; buildings by means of Larson and Greither of Hopf orders over valuation earrings; ramification standards of Byott and Greither for the linked order of the valuation ring of an extension of neighborhood fields to be Hopf order; the Galois module constitution of wildly ramified cyclic extensions of neighborhood fields of measure $p$ and $p^2$; and Kummer conception of formal teams. past a common history in graduate-level algebra, a few chapters think an acquaintance with a few algebraic quantity concept. From there, this exposition serves as an exceptional source and motivation for extra paintings within the box.
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