finite extension sentence in Hindi
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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:
- *PM : a finite extension of fields is an algebraic extension, id = 4725-- WP guess : a finite extension of fields is an algebraic extension-- Status:
- Hello, I'm trying to do the following homework problem : Let L / K be finite extensions of the p-adic rationals \ mathbb { Q } _ p.
- Let the discrete valuation ring " R " be the ring of formal power series over " K " whose coefficients generate a finite extension of " k ".
- Let k'be a finite extension of " k " containing all " q "-th roots of coefficients of finitely many rational functions that generate " L ".
- 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.
- 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 !.
- 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.
- 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.
- 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.
- It's easier not to work with the field of all algebraic numbers, but with finite extensions of the rationals, e . g . the number field generated by some given algebraic number u : K = Q ( u ).
- If " N " is a Galois extension of a Hilbertian field, then although " N " need not be Hilbertian itself, Weisseauer's results asserts that any proper finite extension of " N " is Hilbertian.
- Analogously, the group of p-adic numbers \ Q _ p is isomorphic to its dual . ( In fact, any finite extension of \ Q _ p is also self-dual . ) It follows that the adeles are self-dual.
- If " L " is a finite extension of " K " that is separable ( for example, this is automatically satisfied if " K " is finite or has characteristic zero ) then the following property is also equivalent:
- In case " K " and " L " are finite extensions of N, the situation is particularly simple since the tensor product is of finite dimension as an " N "-algebra ( and thus an Artinian ring ).
- The p-adic fields or any finite extension of them are characteristic zero fields, much applied in number theory, that are constructed from rings of characteristic " p " " k ", as " k " ?! ".
- An extension " E " / " F " is also sometimes said to be simply "'finite "'if it is a finite extension; this should not be confused with the fields themselves being finite fields ( fields with finitely many elements ).
- In a sense, a finitely generated extension is a transcendental generalization of a finite extension since, if the generators in " A " are all algebraic, then " F " ( " A " ) is a finite extension of " F ".
- In a sense, a finitely generated extension is a transcendental generalization of a finite extension since, if the generators in " A " are all algebraic, then " F " ( " A " ) is a finite extension of " F ".
- Let be a number field ( i . e ., a finite extension of \ mathbb Q, the set of rational numbers ), in other words, K = \ mathbb { Q } ( \ theta ) for some algebraic number \ theta \ in \ mathbb { C } by the primitive element theorem.
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