_{1}

^{*}

The notion of preordering, which is a generalization of the notion of ordering, has been introduced by Serre. On the other hand, the notion of round quadratic forms has been introduced by Witt. Based on these ideas, it is here shown that 1) a field **F** is formally real n-pythagorean iff the nth radical, R_{n}F is a preordering (Theorem 2), and 2) a field **F** is n-pythagorean iff for any n-fold Pfister form **ρ**. There exists an odd integer l(**>**1) such that l×**ρ**** **is a round quadratic form (Theorem 8). By considering upper bounds for the number of squares on Pfister’s interpretation, these results finally lead to the main result (Theorem 10) such that the generalization of pythagorean fields coincides with the generalization of Hilbert’s 17th Problem.

In the latter half of the twentieth century, a consideration for the generalization of pythagorean fields has been made by many researchers, e.g., Elman and Lam [

Throughout the paper, let be a field of characteristic different from 2 and be the multiplicative group of. A field is said to be pythagorean if. For a quadratic form over, we put and. Witt [

The class of fields with the following property has been proposed by Elman and Lam [

: Any torsion n-fold Pfister form over F is hyperbolic.

Furthermore, they made a hypothesis that if a field satisfies the property, then the ideal is torsionfree, where is the ideal of even dimensional forms in the Witt ring. Szymiczek [

We denote by the set of n-fold Pfister forms over and by the nth radical of, which is given by. This radical defined by Yucas [

Later, Koziol [

Pythagorean fields are -pythagorean and the class of 1-pythagorean fields is the same as the class of quasipythagorean fields defined by Kijima and Nishi [

On the other hand, a generalization of Hilbert’s 17th Problem has been accomplished by Artin [

.

Furthermore, he showed implicitly in ([

Unexplained notation and terminology refer to [12,13].

Pfister [

Theorem 1. Let be the rational function field in n variables over a real closed field R and be an element of. Then the following statements are equivalent:

1) for all where is defined.

2) holds.

3) is a totally positive element.

We shall prove some results by use of the notions of preorderings (Serre [

Theorem 2. (([

1) F is n-pythagorean.

2) holds for all.

In particular, if F is formally real, these statements are further equivalent to the condition.

3) The nth radical is a preordering.

If a field is n-pythagorean, then . Thus, the following can be obtained.

Corollary 3. (cf. [

Remark 4. Corollary 3 shows a generalization of Hilbert’s 17th Problem. The notion of preordering and nth radical play an important role for this Problem. A typical example of n-pythagorean field is a field of transcendence degree n over a real closed field. Many examples of n-pythagorean fields are known. For example, n-Hilbert fields are so in [

Next, we shall discuss about the generalization of pythagorean fields. The following result is well-known.

Theorem 5. (cf. [

1) The form is round.

2) is pythagorean.

In particular, if the form is anisotropic, then is a formally real field.

Proposition 6. ([16, Proposition 3]). Let be an n-fold Pfister form over F and l be an odd integer. If is a round quadratic form, then holds.

Proof. For any, it is sufficient to show that. The round form means that. Since is an odd integer, we put for some integer. Hence it follows that. On the other hand, since is a Pfister form, holds and then. Thus follows from Witt’s Cancellation Theorem. This implies that.

Corollary 7. ([16, Proposition 3]). If there exist an integer and an odd integer such that the form is anisotropic round, then F is formally real.

Proof. Since the form is round, it follows from Proposition 6 that. If a field is non-real, then. This contradicts the assumption that is anisotropic.

As a characterization of an n-pythagorean property, the following generalization of pythagorean fields can be presented.

Theorem 8. ([

1) F is n-pythagorean.

2) For any, there exists an odd integer such that is a round quadratic form.

Proof. (1) => (2): If a field is n-pythagorean, then is a round quadratic form for any, any positive integer and any.

(2) => (1): This follows from Theorem 2 and Proposition 6.

Theorem 9. The n-pythagorean field is the generalization of pythagorean field and the Pythagoras number of this field is at most.

Proof. If a field is m-pythagorean, then is (m + 1)-pythagorean. Thus, it follows from Theorem 5 and Theorem 8.

Finally, the main result of this paper has been established as follows.

Theorem 10. The generalization of pythagorean fields coincides with the generalization of Hilbert’s 17th Problem.

Proof. If a field is non-real, then has no ordering and moreover holds. Therefore, Hilbert’s 17th Problem results in a problem that if a field is non-real, then does an equality hold? Thus, the required result can be established by use of Corollary 3 and Theorem 9.

Incidentally, the notion of round quadratic forms is connected with the torsion-freeness of the ideal. We shall extend Proposition 2.3 in [

Proposition 11. ([17, Proposition 3.1]). Let be an integer. If F is an n-pythagorean field, then the following statements hold.

1), where is the maximal torsion subgroup of.

2) is torsion-free.

Proof. 1) For any element of, there exists an element of such that. By ([

in. Because of

, the Pfister form is universal and round. Hence and in.

2) For any element of, it is sufficient to show that p = 0. Now there exists an element of such that. Since, it follows from 1) that. Therefore and then for some element from 1). Since is an element of, it follows from ([

Remark 12. In case of 1-pythagorean fields, statements 1) and 2) of Proposition 11 are equivalent (see Remark 2.4 in [

Becker [

The author would like to express his deep appreciation to the late Professor M. Nishi for leading to the quadratic form theory. Also, he is very grateful to Professor S. Kageyama and the late Dr. T. Iwakami for their valuable advices.