Quotient Ring of Cauchy Sequences is Normed 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.
For all $\sequence {x_n} \in \CC$, let $\eqclass {x_n} {}$ denote the left coset $\sequence {x_n} + \NN$
Let $\norm {\, \cdot \,}_1: \CC \,\big / \NN \to \R_{\ge 0}$ be defined by:
- $\ds \forall \eqclass {x_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} }_1 = \lim_{n \mathop \to \infty} \norm {x_n}$
Then $\struct {\CC \,\big / \NN, \norm {\, \cdot \,}_1 }$ is a valued field.
Proof
By Quotient Ring of Cauchy Sequences is Normed Division Ring then $\CC \,\big / \NN$ is a normed division ring.
By Corollary to Quotient Ring of Cauchy Sequences is Normed Division Ring then $\CC \,\big / \NN$ is a field.
The result follows.
$\blacksquare$
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 $1.21$