Quotient Ring of Cauchy Sequences is Division Ring/Corollary 1
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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 $1.19$