place of field

Let $F$ be a field and $\mathrm{\infty }$ an element not belonging to $F$.  The mapping

$$\phi :k\to F\cup \{\mathrm{\infty }\},$$

where $k$ is a field, is called a place of the field $k$, if it satisfies the following conditions.

• The preimage  ${\phi }^{-1}(F)=𝔬$  is a subring of $k$.

• The restriction  ${\phi |}_{𝔬}$  is a ring homomorphism from $𝔬$ to $F$.

• If  $\phi (a)=\mathrm{\infty }$,  then  $\phi (a^{-1})=0$.

It is easy to see that the subring $𝔬$ of the field $k$ is a valuation domain; so any place of a field determines a unique valuation domain in the field.  Conversely, every valuation domain $𝔬$ with field of fractions $k$ determines a place of $k$:

Theorem.

Let $𝔬$ be a valuation domain with field of fractions $k$ and $𝔭$ the maximal ideal of $𝔬$, consisting of the non-units of $𝔬$.  Then the mapping

$$\phi :k\to 𝔬/𝔭\cup \{\mathrm{\infty }\}$$

defined by

$$\phi (x):=\{\begin{array}{cc}x+𝔭\hspace{1em}\mathrm{when}x\in 𝔬,\hfill & \\ \mathrm{\infty }\hspace{1em}\mathrm{when}x\in k\setminus 𝔬,\hfill & \end{array}$$

is a place of the field $k$.

Proof.  Apparently,  ${\phi }^{-1}(𝔬/𝔭)=𝔬$  and the restriction  ${\phi |}_{𝔬}$  is the canonical homomorphism from the ring $𝔬$ onto the residue-class ring $𝔬/𝔭$.  Moreover, if  $\phi (x)=\mathrm{\infty }$,  then $x$ does not belong to the valuation domain $𝔬$ and thus the inverse element $x^{-1}$ must belong to it without being its unit.  Hence $x^{-1}$ belongs to the ideal $𝔭$ which is the kernel of the homomorphism  $\phi |𝔬$.  So we see that  $\phi (x^{-1})=0$.

Title	place of field
Canonical name	PlaceOfField
Date of creation	2013-03-22 14:56:51
Last modified on	2013-03-22 14:56:51
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	16
Author	pahio (2872)
Entry type	Theorem
Classification	msc 13F30
Classification	msc 13A18
Classification	msc 12E99
Synonym	place
Synonym	spot of field
Related topic	KrullValuation
Related topic	ValuationDeterminedByValuationDomain
Related topic	IntegrityCharacterizedByPlaces
Related topic	RamificationOfArchimedeanPlaces
Defines	place of field