purely inseparable sentence in Hindi
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- The absolute Frobenius morphism is a purely inseparable morphism of degree.
- Several equivalent and more concrete definitions for the notion of a purely inseparable extension are known.
- Purely inseparable extensions do occur naturally; for example, they occur in algebraic geometry over fields of prime characteristic.
- It is clear that it is enough to show the assertion when " L " / " K " is either separable or purely inseparable.
- Introduced a variation of Galois theory for purely inseparable extensions of exponent 1, where the Galois groups of field automorphisms in Galois theory are replaced by restricted Lie algebras of derivations.
- The Jacobson Bourbaki theorem implies both the usual Galois correspondence for subfields of a Galois extension, and Jacobson's Galois correspondence for subfields of a purely inseparable extension of exponent at most 1.
- On the other hand, an arbitrary algebraic extension E \ supseteq F may not possess an intermediate extension that is " purely inseparable " over and over which is " separable ".
- In characteristic " p ", an isogeny of degree " p " of abelian varieties must, for their function fields, give either an Artin Schreier extension or a purely inseparable extension.
- The extreme opposite of the concept of separable extension, namely the concept of purely inseparable extension, also occurs quite naturally, as every algebraic extension may be decomposed in a unique way as a purely inseparable extension of separable extension.
- The extreme opposite of the concept of separable extension, namely the concept of purely inseparable extension, also occurs quite naturally, as every algebraic extension may be decomposed in a unique way as a purely inseparable extension of separable extension.
- Remark : The same idea in the proof shows that if L / K is a purely inseparable extension ( need not be normal ), then \ operatorname { Spec } B \ to \ operatorname { Spec } A is bijective.
- 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.
- The known proofs of this equality use the fact that if K \ supseteq F is a purely inseparable extension, and if is a separable irreducible polynomial in, then remains irreducible in " K " [ " X " ] ).
- In particular, \ alpha ^ { p } = a and by the property stated in the paragraph directly above, it follows that F [ \ alpha ] \ supseteq F is a non-trivial purely inseparable extension ( in fact, E = F [ \ alpha ], and so E \ supseteq F is automatically a purely inseparable extension ).
- In particular, \ alpha ^ { p } = a and by the property stated in the paragraph directly above, it follows that F [ \ alpha ] \ supseteq F is a non-trivial purely inseparable extension ( in fact, E = F [ \ alpha ], and so E \ supseteq F is automatically a purely inseparable extension ).
- An algebraic extension E \ supset F of fields of non-zero characteristics is a purely inseparable extension if and only if for every \ alpha \ in E \ setminus F, the minimal polynomial of \ alpha over is " not " a separable polynomial, or, equivalently, for every element of, there is a positive integer such that x ^ { p ^ k } \ in F.
- Local uniformization in positive characteristic seems to be much harder . proved local uniformization in all characteristic for surfaces and in characteristics at least 7 for 3-folds, and was able to deduce global resolution of singularities in these cases from this . simplified Abhyankar's long proof . extended Abhyankar's proof of local uniformization of 3-folds to the remaining characteristics 2, 3, and 5 . showed that it is possible to find a local uniformization of any valuation after taking a purely inseparable extension of the function field.
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