*PM : a finite extension of fields is an algebraic extension, id = 4725-- WP guess : a finite extension of fields is an algebraic extension-- Status:
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*PM : a finite extension of fields is an algebraic extension, id = 4725-- WP guess : a finite extension of fields is an algebraic extension-- Status:
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Hello, I'm trying to do the following homework problem : Let L / K be finite extensions of the p-adic rationals \ mathbb { Q } _ p.
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Let the discrete valuation ring " R " be the ring of formal power series over " K " whose coefficients generate a finite extension of " k ".
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Let k'be a finite extension of " k " containing all " q "-th roots of coefficients of finitely many rational functions that generate " L ".
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If " A " is a Dedekind domain whose quotient field is an algebraic number field ( a finite extension of the rationals ) then shows that SK 1 ( " A " ) vanishes.
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This is a product of finite extensions of ! " p ", in 1 1 correspondence with the completions of K for extensions of the " p "-adic metric on !.
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One important consequence of the theorem is that the integral closure of a Dedekind domain " A " in a finite extension of the field of fractions of " A " is again a Dedekind domain.
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A similar construction works using a primitive nontrivial purely inseparable finite extension of an imperfect field in any positive characteristic, the only difference being that the formula for the norm map is a bit more complicated than in the preceding quadratic examples.
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Penrose concluded that whenever there is a cube where all the outgoing ( and ingoing ) light rays are initially converging, the boundary of the future of that region will end after a finite extension, because all the null geodesics will converge.
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