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Why is the Klein $4$-group and $G(E/F)$ equal?

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Why is the Klein $4$-group and $G(E/F)$ equal? $E=mathbbQ(sqrt1-sqrt3,sqrt1+sqrt3)$ and $F=mathbbQ(sqrt3)$.

I understand that $G(E/F)$ has the same structure as the Klein $4$-group and the same dimension. But is this all that is needed to be shown for equality? Or should I also mention that $G(E/F)$ is abelian?

abstract-algebra galois-theory

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asked Apr 1 at 0:58

numericalorangenumericalorange

1,949314

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  Technically they aren't equal, they are isomorphic. Also by dimension, I suspect you mean size. And all you need to show an isomorphism is to show that the structures of the groups are the same.
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  – jgon
  Apr 1 at 1:02
  

  
  
  
  


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  What do you mean same structure as Klein 4? Have you already proven isomorphism? Clarify what you already have vs what you think is left
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  – AHusain
  Apr 1 at 1:03
  

  
  
  
  


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  @jgon Oh really, they are isomorphic only? I see...
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  – numericalorange
  Apr 1 at 1:05
  

  
  
  
  


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  @AHusain I showed the structures and sizes are the same, so they must be isomorphic. But I thought maybe you could prove they are equal with something else...maybe I am wrong and they cannot be equal.
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  – numericalorange
  Apr 1 at 1:06
  

  
  
  
  


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  @numericalorange Well, it's possible(ish), but it depends on what you mean. Perhaps what you're trying to ask is if you can prove that the Galois group corresponds to the normal Klein four subgroup of the symmetric group on the set of elements $pmsqrt1pm sqrt3$ under the canonical embedding. In which case, this is a sensible question. That said, to improve this question, perhaps make your question more precise, and as AHusain has suggested, clarify what you've done and what you'd like to show.
  $endgroup$
  – jgon
  Apr 1 at 1:09
  
  

  
  
  
  

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show 1 more comment

-1

1

$begingroup$

Why is the Klein $4$-group and $G(E/F)$ equal? $E=mathbbQ(sqrt1-sqrt3,sqrt1+sqrt3)$ and $F=mathbbQ(sqrt3)$.

I understand that $G(E/F)$ has the same structure as the Klein $4$-group and the same dimension. But is this all that is needed to be shown for equality? Or should I also mention that $G(E/F)$ is abelian?

abstract-algebra galois-theory

share|cite|improve this question

asked Apr 1 at 0:58

numericalorangenumericalorange

1,949314

$endgroup$

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  Technically they aren't equal, they are isomorphic. Also by dimension, I suspect you mean size. And all you need to show an isomorphism is to show that the structures of the groups are the same.
  $endgroup$
  – jgon
  Apr 1 at 1:02
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  What do you mean same structure as Klein 4? Have you already proven isomorphism? Clarify what you already have vs what you think is left
  $endgroup$
  – AHusain
  Apr 1 at 1:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @jgon Oh really, they are isomorphic only? I see...
  $endgroup$
  – numericalorange
  Apr 1 at 1:05
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @AHusain I showed the structures and sizes are the same, so they must be isomorphic. But I thought maybe you could prove they are equal with something else...maybe I am wrong and they cannot be equal.
  $endgroup$
  – numericalorange
  Apr 1 at 1:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @numericalorange Well, it's possible(ish), but it depends on what you mean. Perhaps what you're trying to ask is if you can prove that the Galois group corresponds to the normal Klein four subgroup of the symmetric group on the set of elements $pmsqrt1pm sqrt3$ under the canonical embedding. In which case, this is a sensible question. That said, to improve this question, perhaps make your question more precise, and as AHusain has suggested, clarify what you've done and what you'd like to show.
  $endgroup$
  – jgon
  Apr 1 at 1:09
  
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

Why is the Klein $4$-group and $G(E/F)$ equal? $E=mathbbQ(sqrt1-sqrt3,sqrt1+sqrt3)$ and $F=mathbbQ(sqrt3)$.

I understand that $G(E/F)$ has the same structure as the Klein $4$-group and the same dimension. But is this all that is needed to be shown for equality? Or should I also mention that $G(E/F)$ is abelian?

abstract-algebra galois-theory

share|cite|improve this question

asked Apr 1 at 0:58

numericalorangenumericalorange

1,949314

$endgroup$

share|cite|improve this question

asked Apr 1 at 0:58

numericalorangenumericalorange

1,949314

share|cite|improve this question

asked Apr 1 at 0:58

numericalorangenumericalorange

1,949314

asked Apr 1 at 0:58

numericalorangenumericalorange

1,949314

asked Apr 1 at 0:58

numericalorangenumericalorange

1,949314