The corestriction of p-symbols

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 $$\\displaylines{ \\roman{cor}_{K(a)/K}\\left(\\frac{f`(a)}{f(a)},sa+t\\right)_{K(a)}+ \\roman{cor}_{K(c)/K}\\left(\\frac{p`(c)}{p(c)},sc+t\\right)_{K(c)}\\hfill\\cr \\hfill{}=\\roman{cor}_{K(a)/K}[s,f(a))_{K(a)}=\\roman{cor}_{K(c)/K} [s,p(c))_{K(c)}.\\cr}$$\n\n Theorem 4: With the same notations as in Theorem 3, one has $\\roman{cor} _{K(a)/K}[s^pa+t,f(a))_{K(a)}=\\roman{cor}_{K(c)/K}[s^pc+t,p(c))_{K(c)}$. \n 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.