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| − | An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u0953001.png" /> of a discrete valuation ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u0953002.png" /> (cf. [[Discretely-normed ring|Discretely-normed ring]]) with prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u0953003.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u0953004.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u0953005.png" /> are two uniformizing elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u0953006.png" />, then the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u0953007.png" /> is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u0953008.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u0953009.png" /> be a system of representatives in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u09530010.png" /> for the elements of the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u09530011.png" />. Then any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u09530012.png" /> can be uniquely expressed as a power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u09530013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u09530014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u09530015.png" /> is a uniformizing element. If the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u09530016.png" /> is complete relative to the discrete valuation, then any power series of the above form represents an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u09530017.png" />. | + | {{TEX|done}} |
| | + | An element $\pi$ of a discrete valuation ring $A$ (cf. [[Discretely-normed ring|Discretely-normed ring]]) with prime ideal $\mathfrak p$ such that $\mathfrak p=A\pi$. If $\pi_1,\pi_2$ are two uniformizing elements in $A$, then the element $\pi_1\pi_2^{-1}$ is invertible in $A$. Let $R$ be a system of representatives in $A$ for the elements of the residue field $A/\mathfrak p$. Then any element $a\in A$ can be uniquely expressed as a power series $\sum_{i=0}^\infty r_i\pi^i$, where $r_i\in R$ and $\pi$ is a uniformizing element. If the ring $A$ is complete relative to the discrete valuation, then any power series of the above form represents an element $a\in A$. |
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| − | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u09530018.png" /> is the local ring of functions at a simple point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u09530019.png" /> of an [[Algebraic curve|algebraic curve]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u09530020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u09530021.png" /> is a uniformizing element if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u09530022.png" /> has a zero of order one at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095300/u09530023.png" />. | + | If $A$ is the local ring of functions at a simple point $x$ of an [[Algebraic curve|algebraic curve]] $X$, then $\pi$ is a uniformizing element if and only if $\pi$ has a zero of order one at $x$. |
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Latest revision as of 13:16, 14 September 2014
An element $\pi$ of a discrete valuation ring $A$ (cf. Discretely-normed ring) with prime ideal $\mathfrak p$ such that $\mathfrak p=A\pi$. If $\pi_1,\pi_2$ are two uniformizing elements in $A$, then the element $\pi_1\pi_2^{-1}$ is invertible in $A$. Let $R$ be a system of representatives in $A$ for the elements of the residue field $A/\mathfrak p$. Then any element $a\in A$ can be uniquely expressed as a power series $\sum_{i=0}^\infty r_i\pi^i$, where $r_i\in R$ and $\pi$ is a uniformizing element. If the ring $A$ is complete relative to the discrete valuation, then any power series of the above form represents an element $a\in A$.
If $A$ is the local ring of functions at a simple point $x$ of an algebraic curve $X$, then $\pi$ is a uniformizing element if and only if $\pi$ has a zero of order one at $x$.
References
| [a1] | E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9 |
How to Cite This Entry:
Uniformizing element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniformizing_element&oldid=33282
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article