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Title: The corestriction of p-symbols
Authors: Chu, Huah
Kang, Ming Chang
Tan, Eng-Tjioe
Contributors: 應數系
Date: 1988
Issue Date: 2018-09-25 16:23:03 (UTC+8)
Abstract: Let $K$ be a field of characteristic $p>0$. For any $a,b\in K$, $b\not=0$ the $p$-symbol $[a,b)_K$ denotes the similarity class in $\roman{Br}(K)$ of the central simple $p$-algebra: $\bigoplus_{0\leq i,j\leq p-1}Kx^iy^j$, $x^p-x=a$, $y^p=b$, $yx=(x+1)y$; and for any $a,b\in K$, $(a,b)_K$ denotes the class of: $\bigoplus_{0\leq i,j\leq p-1}Kx^iy^j$, $x^p=a$, $y^p=b$, $yx=xy+1$. The following reciprocity laws for the corestriction of the above $p$-symbols are proven. Theorem 3: Let $K(a)$ and $K(c)$ be any finite separable field extensions of $K$, $p(X)$ and $f(X)$ the irreducible polynomials of $a$ and $c$ over $K$, respectively. If $p(X)$ and $f(X)$ are distinct polynomials, for any $s,t\in K$ one has $$\displaylines{ \roman{cor}_{K(a)/K}\left(\frac{f'(a)}{f(a)},sa+t\right)_{K(a)}+ \roman{cor}_{K(c)/K}\left(\frac{p'(c)}{p(c)},sc+t\right)_{K(c)}\hfill\cr \hfill{}=\roman{cor}_{K(a)/K}[s,f(a))_{K(a)}=\roman{cor}_{K(c)/K} [s,p(c))_{K(c)}.\cr}$$

Theorem 4: With the same notations as in Theorem 3, one has $\roman{cor} _{K(a)/K}[s^pa+t,f(a))_{K(a)}=\roman{cor}_{K(c)/K}[s^pc+t,p(c))_{K(c)}$.
These results extend the reciprocity laws of Rosset and Tate for the corestriction of Milnor functions and of P. Mammone [same journal 14 (1986), no. 3, 517–529; MR0823352] for the corestriction of $p$-symbols. Mammone's reciprocity law concerned the multiplicative part of the $p$-symbol, i.e., the second argument. The above result also allows elements to appear in the first variable.
Relation: Communications in Algebra, 16(4), 735-741
AMS MathSciNet:MR932631
Data Type: article
DOI 連結:
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