Definition:Residue Division Ring Induced by Non-Archimedean Norm

Definition

Let $\struct {R, \norm {\,\cdot\,}}$ be a non-Archimedean normed division ring.

Let $\OO$ be the valuation ring induced by the [[[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]] $\norm {\,\cdot\,}$.

Let $\PP$ be the valuation ideal induced by the [[[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]] $\norm {\,\cdot\,}$.

The residue division ring induced by the norm $\norm {\,\cdot\,}$ is the quotient ring $\OO / \PP$.

If $R$ is a field then the quotient ring $\OO / \PP$ is called the residue field induced by the norm $\norm {\,\cdot\,}$.

Also see

• Definition:Valuation Ring Induced by Non-Archimedean Norm

• Valuation Ring of Non-Archimedean Division Ring is Subring, where it is shown that the valuation ring induced by the norm $\norm {\,\cdot\,}$ is a subring of $R$.

• Definition:Valuation Ideal Induced by Non-Archimedean Norm

• Valuation Ideal is Maximal Ideal of Induced Valuation Ring, where it is shown that the valuation ideal is an ideal such that the quotient ring $\OO / \PP$ is a division ring.

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.4$ Algebra: Definition $2.4.2$