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Valuations of polynomial rings Macauley, Ronald Alvin
Abstract
If R is a field on which all (non-archimedean) valuations are known, then all valuations on R[x], where x is transcendental over R , are also known. Ostrowski described such valuations of R[x] by means of pseudo-convergent sequences in the algebraic completion of A of R . MacLane later showed that if all valuations of R are discrete, then any valuation V of R [x] can be represented by certain "key" polynomials in R [x]. The present paper exhibits the connection between these two treatments. This is achieved by first determining keys for the valuation which a pseudo-convergent sequence defines on A[x], and then relating these keys to those for V .
Item Metadata
| Title |
Valuations of polynomial rings
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| Creator | |
| Publisher |
University of British Columbia
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| Date Issued |
1951
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| Description |
If R is a field on which all (non-archimedean) valuations are known, then all valuations on R[x], where x is transcendental over R , are also known. Ostrowski described such valuations of R[x] by means of pseudo-convergent sequences in the algebraic completion of A of R . MacLane later showed that if all valuations of R are discrete, then any valuation V of R [x] can be represented by certain "key" polynomials in R [x]. The present paper exhibits the connection between these two treatments. This is achieved by first determining keys for the valuation which a pseudo-convergent sequence defines on A[x], and then relating these keys to those for V .
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| Genre | |
| Type | |
| Language |
eng
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| Date Available |
2012-03-10
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| Provider |
Vancouver : University of British Columbia Library
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| Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
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| DOI |
10.14288/1.0080629
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| URI | |
| Degree | |
| Program | |
| Affiliation | |
| Degree Grantor |
University of British Columbia
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| Campus | |
| Scholarly Level |
Graduate
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| Aggregated Source Repository |
DSpace
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Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.