On the Generalization of Hilbert ’ s 17 th Problem and Pythagorean Fields

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, is a preordering (Theorem 2), and 2) a field F Rn F is n-pythagorean iff for any n-fold Pfister form  . There exists an odd integer   1 l  such that l   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.

Throughout the paper, let F be a field of characteristic different from 2 and F  be the multiplicative group of F .A field F is said to be . For a quadratic form  over F , we put and . Witt [8] defined a round quadratic form  as Recall that Pfister forms are round ( [8], Satz 4. (c)).
The class of fields with the following property n A has been proposed by Elman and Lam [1]: n A : Any torsion n-fold Pfister form over F is hyperbolic.
Furthermore, they made a hypothesis that if a field F satisfies the property n A , then the ideal n I F is torsionfree, where IF is the ideal of even dimensional forms in the Witt ring .Szymiczek [5] replaced this hypothesis with a problem of rigid elements that if , then 1  , and had studied this problem for amenable fields, linked fields, abstract Witt rings of elementary type, and so on.When a field A , it is clear that F also satisfies the property .This radical defined by Yucas [6] shows a generalization of Kaplansky's radical Later, Koziol [3] has proposed the class of npythagorean fields with the following property as every n-fold Pfister form represents all sums of squares over R F R F [9].
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 [7].In fact, the class of n-pythagorean fields is the same as the class of fields which satisfy the property On the other hand, a generalization of Hilbert's 17th Problem has been accomplished by Artin [10].Later, an interpretation of this generalization has been made by Pfister [11], who has proposed the class of -fields with the following property as for any Unexplained notation and terminology refer to [12,13].

Preorderings and Round Quadratic Forms
Pfister [11] has derived upper bounds for the number of squares on Hilbert's 17th Problem.Hence, the following can be shown by results of Artin [10] and Pfister ([11], chapter 6, Corollary 3.4).Theorem 1.Let be the rational function field in n variables over a real closed field R and be an element of F  .Then the following statements are equivalent: 1) for all , , , We shall prove some results by use of the notions of preorderings (Serre [14]) and round quadratic forms.By Proposition 2.3 in [3] and Lemma 3.1 in [15], the following can be shown.
3) The nth radical is a preordering.
. Thus, the following can be obtained.
Corollary 3. (cf.[13], Corollary 11.4.11).For any formally real n-pythagorean field F, every totally positive element of F is a sum of squares. 2 n 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 [4].Also, Kijima [18] has constructed many such examples by use of some results of Kula [19].
Next, we shall discuss about the generalization of pythagorean fields.The following result is well-known.
Theorem 5. (cf.[8], Satz 3. (g)).Let F be a field and l be an odd integer.Then the following statements are equivalent: 1) The form : l    1 is round.2) F is pythagorean.In particular, if the form  is anisotropic, then F 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
field F 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. ( [16], Proposition 3).For a field F, the following statements are equivalent: 1) F is n-pythagorean.Theorem 9.The n-pythagorean field is the generali- zation of pythagorean field and the Pythagoras number of this field is at most . 2 n Proof.If a field F is m-pythagorean, then F 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 F is non-real, then F has no ordering and moreover holds.Therefore, Hilbert's 17th Problem results in a problem that if a field 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 n I F .We shall extend Proposition 2.3 in [7] to an n- pythagorean field. , , it is sufficient to show that p = 0. Now there exists an element of q 2 I F such that .Since , it follows from 1) that .Therefore t and then for some element from 1).Since is an element of Remark 12.In case of 1-pythagorean fields, statements 1) and 2) of Proposition 11 are equivalent (see Remark 2.4 in [7]).As a characterization of 1-pythagorean fields, Corollary 4.4 of Krawczyk [21] is beautiful and can be extended to -pythagorean fields.This will be given in the forthcoming paper [22].n

Concluding Remark
Becker [2] has used the terminology of n-pythagorean fields F as 2 A  defined by Elman and Lam [1], the following name shall be recommended as Hilbert-Pythagoras field of level n.
quadratic form.Proof.(1)  (2): If a field F is n-pythagorean, then l   is a round quadratic form for any m  (1): This follows from Theorem 2 and Proposition 6.