Quotient Ring of Cauchy Sequences is Division Ring/Corollary 1

Theorem

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

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

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

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

Proof

By Quotient Ring of Cauchy Sequences is Division Ring then $\CC \,\big / \NN$ is a division ring.

By Corollary to Cauchy Sequences form Ring with Unity then $\CC$ is a commutative ring with unity.

By Quotient Ring of Commutative Ring is Commutative then $\CC \,\big / \NN$ is a commutative division ring, that is, a field.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions
• 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.3$: Construction of the completion of a normed field, Theorem $1.19$