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| Title: | The corestriction of p-symbols |
| Authors: | Chu, Huah
Kang, Ming Chang
陳永秋
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 $a,b\\in K$, $b\
ot=0$ 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: | http://dx.doi.org/10.1080/00927878808823599 |
| DOI: | 10.1080/00927878808823599 |
| Appears in Collections: | [應用數學系] 期刊論文
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