Archimedean and Euclidean implies trivial automorphism group - Revision history

Vipul: moved Archimedean and quadratically closed implies trivial automorphism group to Archimedean and Euclidean implies trivial automorphism group

2009-05-14T22:57:37Z

moved Archimedean and quadratically closed implies trivial automorphism group to Archimedean and Euclidean implies trivial automorphism group

← Older revision	Revision as of 22:57, 14 May 2009
(No difference)

Vipul at 22:57, 14 May 2009

2009-05-14T22:57:23Z

← Older revision		Revision as of 22:57, 14 May 2009
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	{{field property implication\|		{{field property implication\|
	stronger = ~~quadratically closed~~ Archimedean field\|		stronger = Euclidean Archimedean field\|
	weaker = field with trivial automorphism group}}		weaker = field with trivial automorphism group}}

	==Statement==		==Statement==

	Suppose <math>K</math> is a [[~~quadratically closed~~ Archimedean field]]: a field that is both [[fact about::~~quadratically closed~~ field\|~~quadratically closed~~]] and [[fact about::Archimedean field\|Archimedean]]. In other words, there exists a total ordering <math>\le</math> on <math>K</math> making <math>K</math> into an [[ordered field]], such that <math>0 \le a</math> if and only if <math>a</math> is a square. Then, the [[automorphism group]] of <math>K</math> is trivial.		Suppose <math>K</math> is a [[Euclidean Archimedean field]]: a field that is both [[fact about::Euclidean field\|Euclidean]] and [[fact about::Archimedean field\|Archimedean]]. In other words, there exists a total ordering <math>\le</math> on <math>K</math> making <math>K</math> into an [[ordered field]], such that <math>0 \le a</math> if and only if <math>a</math> is a square. Then, the [[automorphism group]] of <math>K</math> is trivial.

	==Proof==		==Proof==

Vipul: Created page with '{{field property implication| stronger = quadratically closed Archimedean field| weaker = field with trivial automorphism group}} ==Statement== Suppose $K$ is a [[qu...'

2009-05-14T22:56:51Z

Created page with '{{field property implication| stronger = quadratically closed Archimedean field| weaker = field with trivial automorphism group}} ==Statement== Suppose <math>K</math> is a [[qu...'

New page

{{field property implication|
stronger = quadratically closed Archimedean field|
weaker = field with trivial automorphism group}}

==Statement==

Suppose <math>K</math> is a [[quadratically closed Archimedean field]]: a field that is both [[fact about::quadratically closed field|quadratically closed]] and [[fact about::Archimedean field|Archimedean]]. In other words, there exists a total ordering <math>\le</math> on <math>K</math> making <math>K</math> into an [[ordered field]], such that <math>0 \le a</math> if and only if <math>a</math> is a square. Then, the [[automorphism group]] of <math>K</math> is trivial.

==Proof==

===Proof outline===

# Any field automorphism of <math>K</math> must send squares to squares.
# Any field automorphism of <math>K</math> is also an automorphism of <math>K</math> as an ordered field, i.e., it preserves the total ordering.
# We now use the fact that the field automorphism fixes every rational number, combined with the fact that the rational numbers are dense in <math>K</math>.

Archimedean and Euclidean implies trivial automorphism group - Revision history

Vipul: moved Archimedean and quadratically closed implies trivial automorphism group to Archimedean and Euclidean implies trivial automorphism group

Vipul at 22:57, 14 May 2009

Vipul: Created page with '{{field property implication| stronger = quadratically closed Archimedean field| weaker = field with trivial automorphism group}} ==Statement== Suppose K is a [[qu...'

Vipul: Created page with '{{field property implication| stronger = quadratically closed Archimedean field| weaker = field with trivial automorphism group}} ==Statement== Suppose $K$ is a [[qu...'