# 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:

$$\infty \cdot \infty = \infty ,$$

$$c + \infty = \infty + c = \infty ,\ c \in L,$$

$$c \cdot \infty = \infty \cdot c = \infty ,\ c \in L,\ c \neq 0,$$

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)