On extensions generated by roots of lifting polynomials
On extensions generated by roots of lifting polynomials
复制标题
关于提升多项式的根生成的扩展
DOI：
10.1112/s0025579300016107
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发表时间：
2002
期刊：
影响因子：
0.8
通讯作者：
S. K. Khanduja
中科院分区：
文献类型：
--
作者：
S. Bhatia;S. K. Khanduja
设v是域K的任意秩的亨塞利值及其对K有值群的固定代数闭包的唯一延伸。对于的任意子域L，设R (L)表示约束于L得到的赋值的剩余域。利用v的赋值环在它的残域R (K)上的正则同态，可以提取任何系数在R (K)内的单不可约多项式，得到系数在K内的单不可约多项式。为了推广这一概念，Popescu和Zaharescu引入了关于(K, v)-极小对（α, δ）的提升的概念。在通常的提升情况下，一个给定的属于R (K (α))[y]的单不可约多项式Q (y)会产生K上的几个单不可约多项式，这些多项式是通过对固定的(K, v)-最小对（α, δ）的提升而得到的。如果F， f1是两个这样的提升多项式，其系数在K中分别有根θ， θ 1，那么本文证明了当（K, v）是一个驯服域时，K （θ）和K （θ 1）确实是K -同构的。
Let v be a Henselian valuation of any rank of a field K and its unique prolongation to a fixed algebraic closure of K having value group . For any subfield L of , let R ( L ) denote the residue field of the valuation obtained by restricting to L . Using the canonical homomorphism from the valuation ring of v onto its residue field R ( K ), one can lift any monic irreducible polynomial with coefficients in R ( K ) to yield a monic irreducible polynomial with coefficients in K . In an attempt to generalize this concept, Popescu and Zaharescu introduced the notion of lifting with respect to a ( K, v )-minimal pair ( α, δ ) belonging to × . As in the case of usual lifting, a given monic irreducible polynomial Q ( y ) belonging to R ( K ( α ))[ y ] gives rise to several monic irreducible polynomials over K which are obtained by lifting with respect to a fixed ( K, v )-minimal pair (α, δ). If F , F 1 are two such lifted polynomials with coefficients in K having roots θ, θ 1 , respectively, then it is proved in the present paper that in case ( K, v ) is a tame field, it is shown that K ( θ ) and K ( θ 1 ) are indeed K -isomorphic.