Isomorphic Fields example

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I am trying to show $mathbb Q(2+sqrt2)$ is isomorphic to $mathbb Q(sqrt2)$. Specifically, I have difficulty showing the first is contained in the latter. I must be missing something easy but I cannot seem to find a similar question that has been asked before.






linear-algebra abstract-algebra







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    up vote
    2
    down vote

    favorite












    I am trying to show $mathbb Q(2+sqrt2)$ is isomorphic to $mathbb Q(sqrt2)$. Specifically, I have difficulty showing the first is contained in the latter. I must be missing something easy but I cannot seem to find a similar question that has been asked before.






    linear-algebra abstract-algebra







    share|cite|improve this question

























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      I am trying to show $mathbb Q(2+sqrt2)$ is isomorphic to $mathbb Q(sqrt2)$. Specifically, I have difficulty showing the first is contained in the latter. I must be missing something easy but I cannot seem to find a similar question that has been asked before.






      linear-algebra abstract-algebra







      share|cite|improve this question















      I am trying to show $mathbb Q(2+sqrt2)$ is isomorphic to $mathbb Q(sqrt2)$. Specifically, I have difficulty showing the first is contained in the latter. I must be missing something easy but I cannot seem to find a similar question that has been asked before.






      linear-algebra abstract-algebra




      linear-algebra abstract-algebra






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      share|cite|improve this question













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      edited Sep 1 at 10:15









      mathcounterexamples.net

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      asked Sep 1 at 10:02









      Homaniac

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          If $alpha$ is algebraic of degree two over $mathbb Q$, then $mathbb Q(alpha) =a+balpha colon (a,b) in mathbb Q$. From there it is easy to prove that $mathbb Q(2+sqrt2) =mathbb Q(sqrt2)$.






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          • To prove the two sets are equal, I thought you have to show there are subsets of each other?
            – Homaniac
            Sep 1 at 14:11










          • Yes. Can you try to do it from the definition of $mathbb Q(alpha)$ I provided in the answer?
            – mathcounterexamples.net
            Sep 1 at 14:13










          • Can we take take b =0 to show they are equal sets?
            – Homaniac
            Sep 1 at 14:16










          • Take any element of $mathbb Q(2+sqrt2)$, try to write it as an element of $mathbb Q(sqrt2)$ and vice versa.
            – mathcounterexamples.net
            Sep 1 at 14:18










          • Does $mathbb Q(2+sqrt2) =a+2b+bsqrt2 colon (a,b) in mathbb Q$? So that $a+2b in mathbb Q$ and $b in mathbb Q$ implies the desired result
            – Homaniac
            Sep 1 at 14:23











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          1 Answer
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          active

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          up vote
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          If $alpha$ is algebraic of degree two over $mathbb Q$, then $mathbb Q(alpha) =a+balpha colon (a,b) in mathbb Q$. From there it is easy to prove that $mathbb Q(2+sqrt2) =mathbb Q(sqrt2)$.






          share|cite|improve this answer




















          • To prove the two sets are equal, I thought you have to show there are subsets of each other?
            – Homaniac
            Sep 1 at 14:11










          • Yes. Can you try to do it from the definition of $mathbb Q(alpha)$ I provided in the answer?
            – mathcounterexamples.net
            Sep 1 at 14:13










          • Can we take take b =0 to show they are equal sets?
            – Homaniac
            Sep 1 at 14:16










          • Take any element of $mathbb Q(2+sqrt2)$, try to write it as an element of $mathbb Q(sqrt2)$ and vice versa.
            – mathcounterexamples.net
            Sep 1 at 14:18










          • Does $mathbb Q(2+sqrt2) =a+2b+bsqrt2 colon (a,b) in mathbb Q$? So that $a+2b in mathbb Q$ and $b in mathbb Q$ implies the desired result
            – Homaniac
            Sep 1 at 14:23















          up vote
          1
          down vote













          If $alpha$ is algebraic of degree two over $mathbb Q$, then $mathbb Q(alpha) =a+balpha colon (a,b) in mathbb Q$. From there it is easy to prove that $mathbb Q(2+sqrt2) =mathbb Q(sqrt2)$.






          share|cite|improve this answer




















          • To prove the two sets are equal, I thought you have to show there are subsets of each other?
            – Homaniac
            Sep 1 at 14:11










          • Yes. Can you try to do it from the definition of $mathbb Q(alpha)$ I provided in the answer?
            – mathcounterexamples.net
            Sep 1 at 14:13










          • Can we take take b =0 to show they are equal sets?
            – Homaniac
            Sep 1 at 14:16










          • Take any element of $mathbb Q(2+sqrt2)$, try to write it as an element of $mathbb Q(sqrt2)$ and vice versa.
            – mathcounterexamples.net
            Sep 1 at 14:18










          • Does $mathbb Q(2+sqrt2) =a+2b+bsqrt2 colon (a,b) in mathbb Q$? So that $a+2b in mathbb Q$ and $b in mathbb Q$ implies the desired result
            – Homaniac
            Sep 1 at 14:23













          up vote
          1
          down vote










          up vote
          1
          down vote









          If $alpha$ is algebraic of degree two over $mathbb Q$, then $mathbb Q(alpha) =a+balpha colon (a,b) in mathbb Q$. From there it is easy to prove that $mathbb Q(2+sqrt2) =mathbb Q(sqrt2)$.






          share|cite|improve this answer












          If $alpha$ is algebraic of degree two over $mathbb Q$, then $mathbb Q(alpha) =a+balpha colon (a,b) in mathbb Q$. From there it is easy to prove that $mathbb Q(2+sqrt2) =mathbb Q(sqrt2)$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Sep 1 at 10:14









          mathcounterexamples.net

          25.6k21754




          25.6k21754











          • To prove the two sets are equal, I thought you have to show there are subsets of each other?
            – Homaniac
            Sep 1 at 14:11










          • Yes. Can you try to do it from the definition of $mathbb Q(alpha)$ I provided in the answer?
            – mathcounterexamples.net
            Sep 1 at 14:13










          • Can we take take b =0 to show they are equal sets?
            – Homaniac
            Sep 1 at 14:16










          • Take any element of $mathbb Q(2+sqrt2)$, try to write it as an element of $mathbb Q(sqrt2)$ and vice versa.
            – mathcounterexamples.net
            Sep 1 at 14:18










          • Does $mathbb Q(2+sqrt2) =a+2b+bsqrt2 colon (a,b) in mathbb Q$? So that $a+2b in mathbb Q$ and $b in mathbb Q$ implies the desired result
            – Homaniac
            Sep 1 at 14:23

















          • To prove the two sets are equal, I thought you have to show there are subsets of each other?
            – Homaniac
            Sep 1 at 14:11










          • Yes. Can you try to do it from the definition of $mathbb Q(alpha)$ I provided in the answer?
            – mathcounterexamples.net
            Sep 1 at 14:13










          • Can we take take b =0 to show they are equal sets?
            – Homaniac
            Sep 1 at 14:16










          • Take any element of $mathbb Q(2+sqrt2)$, try to write it as an element of $mathbb Q(sqrt2)$ and vice versa.
            – mathcounterexamples.net
            Sep 1 at 14:18










          • Does $mathbb Q(2+sqrt2) =a+2b+bsqrt2 colon (a,b) in mathbb Q$? So that $a+2b in mathbb Q$ and $b in mathbb Q$ implies the desired result
            – Homaniac
            Sep 1 at 14:23
















          To prove the two sets are equal, I thought you have to show there are subsets of each other?
          – Homaniac
          Sep 1 at 14:11




          To prove the two sets are equal, I thought you have to show there are subsets of each other?
          – Homaniac
          Sep 1 at 14:11












          Yes. Can you try to do it from the definition of $mathbb Q(alpha)$ I provided in the answer?
          – mathcounterexamples.net
          Sep 1 at 14:13




          Yes. Can you try to do it from the definition of $mathbb Q(alpha)$ I provided in the answer?
          – mathcounterexamples.net
          Sep 1 at 14:13












          Can we take take b =0 to show they are equal sets?
          – Homaniac
          Sep 1 at 14:16




          Can we take take b =0 to show they are equal sets?
          – Homaniac
          Sep 1 at 14:16












          Take any element of $mathbb Q(2+sqrt2)$, try to write it as an element of $mathbb Q(sqrt2)$ and vice versa.
          – mathcounterexamples.net
          Sep 1 at 14:18




          Take any element of $mathbb Q(2+sqrt2)$, try to write it as an element of $mathbb Q(sqrt2)$ and vice versa.
          – mathcounterexamples.net
          Sep 1 at 14:18












          Does $mathbb Q(2+sqrt2) =a+2b+bsqrt2 colon (a,b) in mathbb Q$? So that $a+2b in mathbb Q$ and $b in mathbb Q$ implies the desired result
          – Homaniac
          Sep 1 at 14:23





          Does $mathbb Q(2+sqrt2) =a+2b+bsqrt2 colon (a,b) in mathbb Q$? So that $a+2b in mathbb Q$ and $b in mathbb Q$ implies the desired result
          – Homaniac
          Sep 1 at 14:23


















           

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