Place of a field

$ 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)

References

[a1]	P.M. Cohn, "Algebra" , 2 , Wiley (1977) pp. Chapt. 9