Quotient Ring of Cauchy Sequences is Division Ring

Theorem

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

Let $\CC$ be the ring of Cauchy sequences over $R$.

Let $\NN$ be the set of null sequences.

Then the quotient ring $\CC / \NN$ is a division ring.

Corollary

Let $\struct {R, \norm {\, \cdot \,} }$ be a valued field.

Then the quotient ring $\CC \,\big / \NN$ is a field.

Proof

By Null Sequences form Maximal Left and Right Ideal then $\NN$ is an ideal of the ring $\CC$ that is also a maximal left ideal.

By Maximal Left and Right Ideal iff Quotient Ring is Division Ring then the quotient ring $\CC / \NN$ is a division ring

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.3$: Construction of the completion of a normed field: Theorem $1.19$