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- W2079202009 abstract "Examples are given for rational function fields that do not satisfy the strong Hasse principle for quadratic forms. One of the most celebrated results in the theory of quadratic forms is the Hasse-Minkowski Theorem. It asserts that two quadratic forms over an algebraic number field are equivalent if and only if they are locally (everywhere) equivalent. This result is generally brought forth as a consequence of the following stronger version of the so-called local-global principle: a quadratic form over a number field is isotropic (i.e. represents zero nontrivially) if and only if it is isotropic everywhere locally; see [BS], [OM]. What is, however, sometimes not recognized is that there are fields which satisfy the weak-but not the strong-Hasse principle. In this note we shall furnish some examples of rational function fields that do not satisfy the strong Hasse principle, and also make a few remarks about the two versions of the Hasse principle. Let F be a field of characteristic different from 2, and S a set of inequivalent valuations on F. The members of S shall be denoted by the p's. We say a quadratic form k over F satisfies the Strong Hasse Principle (SHP) w.r.t. S if p is locally isotropic at each p E S implies 0 is isotropic. (As usual, locally here means with respect to the completion F. of F at p.) The pair (F, S) satisfies SHP if every quadratic form over F satisfies SHP w.r.t. S. Two quadratic forms over F satisfy the Weak Hasse Principle (WHP) w.r.t. S if local equivalence between them at each p E S implies global equivalence. Similarly, the pair (F, S) satisfies WHP if every two quadratic forms over F satisfy WHP w.r.t. S. It is immediate that SHP always implies WHP. For low enough dimensions, the converse is also valid; namely: (i) If WHP holdsforforms with dimension <3 w. r. t. S, then so does SHP. Received by the editors November 3, 1972. AMS (MOS) subject classifications (1970). Primary 1OCO5, 15A63; Secondary 10J05, 14H05." @default.
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- W2079202009 date "1973-03-01" @default.
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- W2079202009 title "On the Hasse principle for quadratic forms" @default.
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- W2079202009 doi "https://doi.org/10.1090/s0002-9939-1973-0311572-3" @default.
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