A visually guided proof of the fundamental theorem of algebra?

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A complex root of a polynomial $P(z)$ is a pair of real numbers $u,v$ that simultaneously make the real part and the imaginary part of $P(z)$ zero.



The zeros of the real part and the imaginary part are given by two curves in the complex plane



$$operatornameRe(P(u +iv)) = 0$$



$$operatornameIm(P(u+iv)) = 0$$



The zeros of the polynomial are the intersection points of these two curves.



For the sake of specifity here are the two curves for $P(z) = z^5 + z^3 + z^2 + z + 1$:



$u^5-6u^3v^2-4v^2u^3+5uv^4+ u^3-v^2u-2uv^2+u^2-v^2+u+1 = 0 $



$v^5+5vu^4-10u^2v^3+3u^2v-v^3+2uv+v=0$



The fundamental theorem of algebra says that such curves always do intersect. A proof of the fundamental theorem might go like this: The curves $operatornameRe(P(z)) = 0$ (red) and $operatornameIm(P(z)) = 0$ (blue) – which are tightly related – are such-and-such, so they must intersect at least once and at most $n$ times (for $n$ the degree of the polynomial).



What can be seen is, that the curves always come in $n$ branches which extend to infinity and for some reason must intersect.



enter image description here$x^2 + x + 1$



enter image description here$x^3 + x^2 + x + 1$



enter image description here$x^4 + x^2 + x + 1$



enter image description here$x^5 + x^3 + x^2 + x + 1$



How could such a proof be spelled out?






abstract-algebra polynomials complex-numbers proof-writing roots







share|cite|improve this question



























    up vote
    4
    down vote

    favorite
    1












    A complex root of a polynomial $P(z)$ is a pair of real numbers $u,v$ that simultaneously make the real part and the imaginary part of $P(z)$ zero.



    The zeros of the real part and the imaginary part are given by two curves in the complex plane



    $$operatornameRe(P(u +iv)) = 0$$



    $$operatornameIm(P(u+iv)) = 0$$



    The zeros of the polynomial are the intersection points of these two curves.



    For the sake of specifity here are the two curves for $P(z) = z^5 + z^3 + z^2 + z + 1$:



    $u^5-6u^3v^2-4v^2u^3+5uv^4+ u^3-v^2u-2uv^2+u^2-v^2+u+1 = 0 $



    $v^5+5vu^4-10u^2v^3+3u^2v-v^3+2uv+v=0$



    The fundamental theorem of algebra says that such curves always do intersect. A proof of the fundamental theorem might go like this: The curves $operatornameRe(P(z)) = 0$ (red) and $operatornameIm(P(z)) = 0$ (blue) – which are tightly related – are such-and-such, so they must intersect at least once and at most $n$ times (for $n$ the degree of the polynomial).



    What can be seen is, that the curves always come in $n$ branches which extend to infinity and for some reason must intersect.



    enter image description here$x^2 + x + 1$



    enter image description here$x^3 + x^2 + x + 1$



    enter image description here$x^4 + x^2 + x + 1$



    enter image description here$x^5 + x^3 + x^2 + x + 1$



    How could such a proof be spelled out?






    abstract-algebra polynomials complex-numbers proof-writing roots







    share|cite|improve this question

























      up vote
      4
      down vote

      favorite
      1









      up vote
      4
      down vote

      favorite
      1






      1





      A complex root of a polynomial $P(z)$ is a pair of real numbers $u,v$ that simultaneously make the real part and the imaginary part of $P(z)$ zero.



      The zeros of the real part and the imaginary part are given by two curves in the complex plane



      $$operatornameRe(P(u +iv)) = 0$$



      $$operatornameIm(P(u+iv)) = 0$$



      The zeros of the polynomial are the intersection points of these two curves.



      For the sake of specifity here are the two curves for $P(z) = z^5 + z^3 + z^2 + z + 1$:



      $u^5-6u^3v^2-4v^2u^3+5uv^4+ u^3-v^2u-2uv^2+u^2-v^2+u+1 = 0 $



      $v^5+5vu^4-10u^2v^3+3u^2v-v^3+2uv+v=0$



      The fundamental theorem of algebra says that such curves always do intersect. A proof of the fundamental theorem might go like this: The curves $operatornameRe(P(z)) = 0$ (red) and $operatornameIm(P(z)) = 0$ (blue) – which are tightly related – are such-and-such, so they must intersect at least once and at most $n$ times (for $n$ the degree of the polynomial).



      What can be seen is, that the curves always come in $n$ branches which extend to infinity and for some reason must intersect.



      enter image description here$x^2 + x + 1$



      enter image description here$x^3 + x^2 + x + 1$



      enter image description here$x^4 + x^2 + x + 1$



      enter image description here$x^5 + x^3 + x^2 + x + 1$



      How could such a proof be spelled out?






      abstract-algebra polynomials complex-numbers proof-writing roots







      share|cite|improve this question















      A complex root of a polynomial $P(z)$ is a pair of real numbers $u,v$ that simultaneously make the real part and the imaginary part of $P(z)$ zero.



      The zeros of the real part and the imaginary part are given by two curves in the complex plane



      $$operatornameRe(P(u +iv)) = 0$$



      $$operatornameIm(P(u+iv)) = 0$$



      The zeros of the polynomial are the intersection points of these two curves.



      For the sake of specifity here are the two curves for $P(z) = z^5 + z^3 + z^2 + z + 1$:



      $u^5-6u^3v^2-4v^2u^3+5uv^4+ u^3-v^2u-2uv^2+u^2-v^2+u+1 = 0 $



      $v^5+5vu^4-10u^2v^3+3u^2v-v^3+2uv+v=0$



      The fundamental theorem of algebra says that such curves always do intersect. A proof of the fundamental theorem might go like this: The curves $operatornameRe(P(z)) = 0$ (red) and $operatornameIm(P(z)) = 0$ (blue) – which are tightly related – are such-and-such, so they must intersect at least once and at most $n$ times (for $n$ the degree of the polynomial).



      What can be seen is, that the curves always come in $n$ branches which extend to infinity and for some reason must intersect.



      enter image description here$x^2 + x + 1$



      enter image description here$x^3 + x^2 + x + 1$



      enter image description here$x^4 + x^2 + x + 1$



      enter image description here$x^5 + x^3 + x^2 + x + 1$



      How could such a proof be spelled out?






      abstract-algebra polynomials complex-numbers proof-writing roots




      abstract-algebra polynomials complex-numbers proof-writing roots






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      edited 48 mins ago









      José Carlos Santos

      130k17106192




      130k17106192










      asked 1 hour ago









      Hans Stricker

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      5,02123882




















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          That idea is basically the approach to the Fundamental Theorem of Algebra taken by Gauss, in his PhD thesis Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse. It's a nice geometrical approach to the theorem, but hard to complete rigorously. I suggest that you read C. F. Gauss’s proofs of the Fundamental Theorem of Algebra, by Harel Cain.






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          • What a nice paper! Thank you very much.
            – Hans Stricker
            51 mins ago










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          up vote
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          accepted










          That idea is basically the approach to the Fundamental Theorem of Algebra taken by Gauss, in his PhD thesis Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse. It's a nice geometrical approach to the theorem, but hard to complete rigorously. I suggest that you read C. F. Gauss’s proofs of the Fundamental Theorem of Algebra, by Harel Cain.






          share|cite|improve this answer




















          • What a nice paper! Thank you very much.
            – Hans Stricker
            51 mins ago














          up vote
          5
          down vote



          accepted










          That idea is basically the approach to the Fundamental Theorem of Algebra taken by Gauss, in his PhD thesis Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse. It's a nice geometrical approach to the theorem, but hard to complete rigorously. I suggest that you read C. F. Gauss’s proofs of the Fundamental Theorem of Algebra, by Harel Cain.






          share|cite|improve this answer




















          • What a nice paper! Thank you very much.
            – Hans Stricker
            51 mins ago












          up vote
          5
          down vote



          accepted







          up vote
          5
          down vote



          accepted






          That idea is basically the approach to the Fundamental Theorem of Algebra taken by Gauss, in his PhD thesis Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse. It's a nice geometrical approach to the theorem, but hard to complete rigorously. I suggest that you read C. F. Gauss’s proofs of the Fundamental Theorem of Algebra, by Harel Cain.






          share|cite|improve this answer












          That idea is basically the approach to the Fundamental Theorem of Algebra taken by Gauss, in his PhD thesis Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse. It's a nice geometrical approach to the theorem, but hard to complete rigorously. I suggest that you read C. F. Gauss’s proofs of the Fundamental Theorem of Algebra, by Harel Cain.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 54 mins ago









          José Carlos Santos

          130k17106192




          130k17106192











          • What a nice paper! Thank you very much.
            – Hans Stricker
            51 mins ago
















          • What a nice paper! Thank you very much.
            – Hans Stricker
            51 mins ago















          What a nice paper! Thank you very much.
          – Hans Stricker
          51 mins ago




          What a nice paper! Thank you very much.
          – Hans Stricker
          51 mins ago

















           

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