Witt ring of field

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Why $W(C)$ is isometric to $mathbbZ_2$ and $W(R)=mathbbZ$ how to get this explicitly? I know we can count $mathbbC$ as even dimensional space over $mathbbR$ so hyperbolic space hence it is $mathbbZ_2$ but I can't get why it is true explicitly? Also for algebraically closed field it is isomorphic to $mathbbZ_2$ please explain this also.




linear-algebra abstract-algebra group-theory




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    Why $W(C)$ is isometric to $mathbbZ_2$ and $W(R)=mathbbZ$ how to get this explicitly? I know we can count $mathbbC$ as even dimensional space over $mathbbR$ so hyperbolic space hence it is $mathbbZ_2$ but I can't get why it is true explicitly? Also for algebraically closed field it is isomorphic to $mathbbZ_2$ please explain this also.




    linear-algebra abstract-algebra group-theory




    share|cite|improve this question
























      up vote
      1
      down vote

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      up vote
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      Why $W(C)$ is isometric to $mathbbZ_2$ and $W(R)=mathbbZ$ how to get this explicitly? I know we can count $mathbbC$ as even dimensional space over $mathbbR$ so hyperbolic space hence it is $mathbbZ_2$ but I can't get why it is true explicitly? Also for algebraically closed field it is isomorphic to $mathbbZ_2$ please explain this also.




      linear-algebra abstract-algebra group-theory




      share|cite|improve this question














      Why $W(C)$ is isometric to $mathbbZ_2$ and $W(R)=mathbbZ$ how to get this explicitly? I know we can count $mathbbC$ as even dimensional space over $mathbbR$ so hyperbolic space hence it is $mathbbZ_2$ but I can't get why it is true explicitly? Also for algebraically closed field it is isomorphic to $mathbbZ_2$ please explain this also.





      linear-algebra abstract-algebra group-theory





      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Aug 16 at 8:39









      Bernard

      111k635103




      111k635103










      asked Aug 16 at 8:26









      Ninja hatori

      149113




      149113




















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          Calculations of the Witt ring are way too demanding for simple Math Stackexchange answer. I'll give you hints and resources.



          For a field $k$ of $textchar(k)neq 2$ the following are equivalent:



          • $k$ does not admit quadratic field extension

          • $W(k)simeqmathbbZ_2$

          You can find the proof in the following paper (Proposition 11):



          https://pdfs.semanticscholar.org/a1c5/a6eb6f12a8dfd0f1f01cf57d6b4fa4233beb.pdf



          Now since algebraically closed fields have no finite extensions then in particular they don't have quadratic field extensions so their Witt ring is $mathbbZ_2$.



          For reals $mathbbR$ read the same paper, Proposition 26. I actually encourage you to read the whole paper.






          share|cite|improve this answer






















          • math.stackexchange.com/questions/2888950/…
            – Ninja hatori
            Aug 22 at 11:19










          • paper doesn't included proof for most of theorems.
            – Ninja hatori
            Aug 22 at 11:19










          Your Answer




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          1 Answer
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          active

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          active

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          up vote
          0
          down vote













          Calculations of the Witt ring are way too demanding for simple Math Stackexchange answer. I'll give you hints and resources.



          For a field $k$ of $textchar(k)neq 2$ the following are equivalent:



          • $k$ does not admit quadratic field extension

          • $W(k)simeqmathbbZ_2$

          You can find the proof in the following paper (Proposition 11):



          https://pdfs.semanticscholar.org/a1c5/a6eb6f12a8dfd0f1f01cf57d6b4fa4233beb.pdf



          Now since algebraically closed fields have no finite extensions then in particular they don't have quadratic field extensions so their Witt ring is $mathbbZ_2$.



          For reals $mathbbR$ read the same paper, Proposition 26. I actually encourage you to read the whole paper.






          share|cite|improve this answer






















          • math.stackexchange.com/questions/2888950/…
            – Ninja hatori
            Aug 22 at 11:19










          • paper doesn't included proof for most of theorems.
            – Ninja hatori
            Aug 22 at 11:19














          up vote
          0
          down vote













          Calculations of the Witt ring are way too demanding for simple Math Stackexchange answer. I'll give you hints and resources.



          For a field $k$ of $textchar(k)neq 2$ the following are equivalent:



          • $k$ does not admit quadratic field extension

          • $W(k)simeqmathbbZ_2$

          You can find the proof in the following paper (Proposition 11):



          https://pdfs.semanticscholar.org/a1c5/a6eb6f12a8dfd0f1f01cf57d6b4fa4233beb.pdf



          Now since algebraically closed fields have no finite extensions then in particular they don't have quadratic field extensions so their Witt ring is $mathbbZ_2$.



          For reals $mathbbR$ read the same paper, Proposition 26. I actually encourage you to read the whole paper.






          share|cite|improve this answer






















          • math.stackexchange.com/questions/2888950/…
            – Ninja hatori
            Aug 22 at 11:19










          • paper doesn't included proof for most of theorems.
            – Ninja hatori
            Aug 22 at 11:19












          up vote
          0
          down vote










          up vote
          0
          down vote









          Calculations of the Witt ring are way too demanding for simple Math Stackexchange answer. I'll give you hints and resources.



          For a field $k$ of $textchar(k)neq 2$ the following are equivalent:



          • $k$ does not admit quadratic field extension

          • $W(k)simeqmathbbZ_2$

          You can find the proof in the following paper (Proposition 11):



          https://pdfs.semanticscholar.org/a1c5/a6eb6f12a8dfd0f1f01cf57d6b4fa4233beb.pdf



          Now since algebraically closed fields have no finite extensions then in particular they don't have quadratic field extensions so their Witt ring is $mathbbZ_2$.



          For reals $mathbbR$ read the same paper, Proposition 26. I actually encourage you to read the whole paper.






          share|cite|improve this answer














          Calculations of the Witt ring are way too demanding for simple Math Stackexchange answer. I'll give you hints and resources.



          For a field $k$ of $textchar(k)neq 2$ the following are equivalent:



          • $k$ does not admit quadratic field extension

          • $W(k)simeqmathbbZ_2$

          You can find the proof in the following paper (Proposition 11):



          https://pdfs.semanticscholar.org/a1c5/a6eb6f12a8dfd0f1f01cf57d6b4fa4233beb.pdf



          Now since algebraically closed fields have no finite extensions then in particular they don't have quadratic field extensions so their Witt ring is $mathbbZ_2$.



          For reals $mathbbR$ read the same paper, Proposition 26. I actually encourage you to read the whole paper.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Aug 17 at 9:11

























          answered Aug 17 at 9:05









          freakish

          8,6971524




          8,6971524











          • math.stackexchange.com/questions/2888950/…
            – Ninja hatori
            Aug 22 at 11:19










          • paper doesn't included proof for most of theorems.
            – Ninja hatori
            Aug 22 at 11:19
















          • math.stackexchange.com/questions/2888950/…
            – Ninja hatori
            Aug 22 at 11:19










          • paper doesn't included proof for most of theorems.
            – Ninja hatori
            Aug 22 at 11:19















          math.stackexchange.com/questions/2888950/…
          – Ninja hatori
          Aug 22 at 11:19




          math.stackexchange.com/questions/2888950/…
          – Ninja hatori
          Aug 22 at 11:19












          paper doesn't included proof for most of theorems.
          – Ninja hatori
          Aug 22 at 11:19




          paper doesn't included proof for most of theorems.
          – Ninja hatori
          Aug 22 at 11:19












           

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