Chain signatures and higher level Witt rings. (English) Zbl 0632.12022
From the author’s introduction: “A field with a higher level ordering has nonisomorphic real closures. Becker showed that in order to describe the isomorphism classes of real closures a whole sequence of higher level orderings is required [E. Becker, “Hereditarily-Pythagorean fields and orderings of higher level”, IMPA, Rio de Janeiro (1978; Zbl 0509.12020), p. 162, theorem 12]. Subsequently, this has been refined and reformulated through the introduction of chains of higher level orderings [J. Harman, Contemp. Math. 8, 141–174 (1982; Zbl 0509.12021)] and chain signatures [the author, J. Reine Angew. Math. 347, 1–20 (1984; Zbl 0531.12017)].
In view of this situation one should expect that if results from the usual theory of ordered fields (level 1) carry over to the case of orderings of higher level then quite different results will also be brought in by the chains of higher level orderings.
The purpose of this paper is to look for such results in the theory of higher level Witt rings developed recently by E. Becker and A. Rosenberg [J. Algebra 92, 477–503 (1985; Zbl 0555.10009)]. To keep things from getting too complicated this will be done only for the case where all levels are powers of 2.”
In view of this situation one should expect that if results from the usual theory of ordered fields (level 1) carry over to the case of orderings of higher level then quite different results will also be brought in by the chains of higher level orderings.
The purpose of this paper is to look for such results in the theory of higher level Witt rings developed recently by E. Becker and A. Rosenberg [J. Algebra 92, 477–503 (1985; Zbl 0555.10009)]. To keep things from getting too complicated this will be done only for the case where all levels are powers of 2.”
Reviewer: Nicolae Radu (Bucureşti)
MSC:
| 12D15 | Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) |
| 11E81 | Algebraic theory of quadratic forms; Witt groups and rings |
| 19G12 | Witt groups of rings |
Keywords:
formally real fields; chain signatures; chains of higher level orderings; higher level Witt ringsReferences:
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