On trace forms of Hilbertian fields.

*(English)*Zbl 0662.10015A quadratic form over a field K is called positive if its signature is nonnegative for every possible ordering of the field K; and it is called algebraic if it is Witt equivalent to the trace form of an algebraic extension L of K. The authors consider the question of when positive forms are algebraic.

Suppose K has the property that whenever a polynomial f(T,X) over K in indeterminates T and X is irreducible as a polynomial in X over K(T), then there exists an element \(\beta\) of K such that f(\(\beta\),X) is an irreducible polynomial in X over K. It is proved that the form \(\delta X^ 2\) is algebraic for any element \(\delta\) of K which is positive for every possible ordering of K. This theorem is combined with a theorem by W. C. Waterhouse [Arch. Math. 47, 229-231 (1986; Zbl 0607.10013)] to give many special cases in which positive forms are algebraic.

Suppose K has the property that whenever a polynomial f(T,X) over K in indeterminates T and X is irreducible as a polynomial in X over K(T), then there exists an element \(\beta\) of K such that f(\(\beta\),X) is an irreducible polynomial in X over K. It is proved that the form \(\delta X^ 2\) is algebraic for any element \(\delta\) of K which is positive for every possible ordering of K. This theorem is combined with a theorem by W. C. Waterhouse [Arch. Math. 47, 229-231 (1986; Zbl 0607.10013)] to give many special cases in which positive forms are algebraic.

Reviewer: H.F.Kreimer

##### MSC:

11E04 | Quadratic forms over general fields |

11E16 | General binary quadratic forms |

12F10 | Separable extensions, Galois theory |