Waste less time on Facebook — follow Brilliant.
×

A Real Injection

Suppose that \((K,\pi,\sigma,1,0,\geq)\) is an ordered field. My question is does there exist a complete ordered field \((K',\pi',\sigma',1',0',\geq')\) and an injective map \(\phi :K \longrightarrow K'\) such that \(\phi(\pi(a,b)) = \pi'(\phi(a),\phi(b)), \phi(\sigma(a,b)) = \sigma'(\phi(a),\phi(b)), a \geq b \Longrightarrow \phi(a) \geq' \phi(b)\) Because if there is then since ever ordered field is of characteristic zero (because \(0<1<2<... \Longleftarrow [1 \neq 0 \Longleftarrow 1 = 1^{2} > 0]\) then \((K,\pi,\sigma,1,0,\geq)\) would be isomorphic to one of the intermediate fields of \(\mathbb{R}|\mathbb{Q}\)

Note by Samuel Queen
4 years, 1 month ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...