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Gelfand--Tornheim theorem
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(Theorem)
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The normed field means a field $K$ having a subfield $R$ isomorphic to $\mathbb{R}$ and satisfying the following: There is a mapping $\|\cdot\|$ from $K$ to the set of non-negative reals such that
- $\|a\| = 0$ iff $a = 0$
- $\|ab\| \leqq \|a\|\cdot\|b\|$
- $\|a+b\| \leqq \|a\|+\|b\|$
- $\|ab\| = |a|\cdot\|b\|$ when $a \in R$ and $b \in K$
Using the Gelfand-Tornheim theorem, it can be shown that the only fields with archimedean valuation are isomorphic to subfields of $\mathbb{C}$ and that the valuation is the usual absolute value (modulus) or some positive power of the absolute value.
- 1
- Emil Artin: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).
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"Gelfand--Tornheim theorem" is owned by pahio.
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Cross-references: power, positive, modulus, valuation, archimedean, iff, mapping, subfield, complex numbers, real numbers, field, isomorphic
This is version 37 of Gelfand--Tornheim theorem, born on 2004-02-26, modified 2008-09-06.
Object id is 5628, canonical name is GelfandTornheimTheorem.
Accessed 4842 times total.
Classification:
| AMS MSC: | 12J05 (Field theory and polynomials :: Topological fields :: Normed fields) |
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Pending Errata and Addenda
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