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[parent] trivial valuation (Definition)

The trivial valuation of a field $ K$ is the Krull valuation $ \vert\cdot\vert$ of $ K$ such that $ \vert\vert = 0$ and $ \vert x\vert = 1$ for other elements $ x$ of $ K$.

Properties

  1. Every field has the trivial valuation.
  2. The trivial valuation is non-archimedean.
  3. The valuation ring of the trivial valuation is the whole field and the corresponding maximal ideal is the zero ideal.
  4. The field is complete with respect to (the metric given by) its trivial valuation.
  5. A finite field has only the trivial valuation. (Let $ a$ be the primitive element of the multiplicative group of the field, which is cyclic. If $ \vert\cdot\vert$ is any valuation of the field, then one must have $ \vert a\vert = 1$ since otherwise $ \vert 1\vert \neq 1$. Consequently, $ \vert x\vert = \vert a^m\vert = \vert a\vert^m = 1^m = 1$ for all non-zero elements $ x$.)
  6. Every algebraic extension of finite fields has only the trivial valuation, but every field of characteristic 0 has non-trivial valuations.



"trivial valuation" is owned by pahio.
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See Also: independence of valuations, Krull valuation


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Cross-references: characteristic, algebraic extension, valuation, multiplicative group of the field, primitive element, finite field, metric, zero ideal, maximal ideal, valuation ring, non-archimedean, Krull valuation, field
There are 5 references to this entry.

This is version 13 of trivial valuation, born on 2004-04-29, modified 2006-12-18.
Object id is 5811, canonical name is TrivialValuation.
Accessed 1846 times total.

Classification:
AMS MSC: 11R99 (Number theory :: Algebraic number theory: global fields :: Miscellaneous)
 12J20 (Field theory and polynomials :: Topological fields :: General valuation theory)
Pending Errata and Addenda
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Discussion
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6. by lance on 2006-12-14 22:35:03
part 6. isn't correct. the algebraic closure of any finite field has no non-trivial valuation. maybe better to say that a field of characterstic 0 has a non trivial valuation?
[ reply | up ]
  • Re: 6. by pahio on 2006-12-15 04:36:54
    • Re: 6. by lance on 2006-12-15 17:54:37
      • Re: 6. by pahio on 2006-12-18 13:03:29

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