Difference between revisions of "Place of a field"
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| − | + | '' $ K $ | |
| + | with values in a field $ L $, | ||
| + | $ L $- | ||
| + | valued place of a field $ K $'' | ||
| − | + | A mapping $ f: K \rightarrow L \cup \{ \infty \} $ | |
| + | satisfying the conditions | ||
| − | + | $$ | |
| + | f ( 1) = 1, | ||
| + | $$ | ||
| + | |||
| + | $$ | ||
| + | f ( a + b) = f ( a) + f ( b), | ||
| + | $$ | ||
| + | |||
| + | $$ | ||
| + | f ( ab) = f ( a) \cdot f ( b) | ||
| + | $$ | ||
(provided that the expressions on the right-hand sides are defined). The following conventions are made: | (provided that the expressions on the right-hand sides are defined). The following conventions are made: | ||
| − | + | $$ | |
| + | \infty \cdot \infty = \infty , | ||
| + | $$ | ||
| − | + | $$ | |
| + | c + \infty = \infty + c = \infty ,\ c \in L, | ||
| + | $$ | ||
| − | + | $$ | |
| + | c \cdot \infty = \infty \cdot c = \infty ,\ c \in L,\ c \neq 0, | ||
| + | $$ | ||
| − | while the expressions | + | while the expressions $ \infty + \infty $, |
| + | $ 0 \cdot \infty $ | ||
| + | and $ \infty \cdot 0 $ | ||
| + | are undefined. | ||
| − | An element | + | An element $ a $ |
| + | in $ K $ | ||
| + | for which $ f ( a) \in L $ | ||
| + | is called finite in the place $ f $; | ||
| + | the set $ A $ | ||
| + | of finite elements is a subring of $ K $, | ||
| + | and the mapping $ f: A \rightarrow L $ | ||
| + | is a ring homomorphism. The ring $ A $ | ||
| + | is a [[Local ring|local ring]], its maximal ideal is $ \mathfrak m = \{ {a \in K } : {f ( a) = 0 } \} $. | ||
| − | A place | + | A place $ f $ |
| + | determines a [[Valuation|valuation]] $ v $ | ||
| + | of $ K $ | ||
| + | with group of values $ K ^ {*} /A ^ {*} $( | ||
| + | where $ K ^ {*} = K \setminus \{ 0 \} $ | ||
| + | and $ A ^ {*} = A \setminus \mathfrak m $ | ||
| + | are, respectively, the groups of invertible elements of $ K $ | ||
| + | and $ A $). | ||
| + | The ring of this valuation is the same as $ A $. | ||
| + | Conversely, any valuation $ v $ | ||
| + | of a field $ K $ | ||
| + | determines a place of $ K $ | ||
| + | with values in the residue class field of $ v $. | ||
| + | Here, the ring of finite elements is the same as the ring of (integers of) the valuation $ v $. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1984)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1984)</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.M. Cohn, "Algebra" , '''2''' , Wiley (1977) pp. Chapt. 9</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.M. Cohn, "Algebra" , '''2''' , Wiley (1977) pp. Chapt. 9</TD></TR></table> | ||
Latest revision as of 08:06, 6 June 2020
$ K $
with values in a field $ L $,
$ L $-
valued place of a field $ K $
A mapping $ f: K \rightarrow L \cup \{ \infty \} $ satisfying the conditions
$$ f ( 1) = 1, $$
$$ f ( a + b) = f ( a) + f ( b), $$
$$ f ( ab) = f ( a) \cdot f ( b) $$
(provided that the expressions on the right-hand sides are defined). The following conventions are made:
$$ \infty \cdot \infty = \infty , $$
$$ c + \infty = \infty + c = \infty ,\ c \in L, $$
$$ c \cdot \infty = \infty \cdot c = \infty ,\ c \in L,\ c \neq 0, $$
while the expressions $ \infty + \infty $, $ 0 \cdot \infty $ and $ \infty \cdot 0 $ are undefined.
An element $ a $ in $ K $ for which $ f ( a) \in L $ is called finite in the place $ f $; the set $ A $ of finite elements is a subring of $ K $, and the mapping $ f: A \rightarrow L $ is a ring homomorphism. The ring $ A $ is a local ring, its maximal ideal is $ \mathfrak m = \{ {a \in K } : {f ( a) = 0 } \} $.
A place $ f $ determines a valuation $ v $ of $ K $ with group of values $ K ^ {*} /A ^ {*} $( where $ K ^ {*} = K \setminus \{ 0 \} $ and $ A ^ {*} = A \setminus \mathfrak m $ are, respectively, the groups of invertible elements of $ K $ and $ A $). The ring of this valuation is the same as $ A $. Conversely, any valuation $ v $ of a field $ K $ determines a place of $ K $ with values in the residue class field of $ v $. Here, the ring of finite elements is the same as the ring of (integers of) the valuation $ v $.
References
| [1] | S. Lang, "Algebra" , Addison-Wesley (1984) |
Comments
References
| [a1] | P.M. Cohn, "Algebra" , 2 , Wiley (1977) pp. Chapt. 9 |
How to Cite This Entry:
Place of a field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Place_of_a_field&oldid=17112
Place of a field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Place_of_a_field&oldid=17112
This article was adapted from an original article by Yu.G. Zarkhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article