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In 1823, Niels Hendrik Abel managed to prove that there is no algebraic formula to solve equations of degree 5
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योगेन्द्र प्रताप सिंह 29. sij
Odgovor korisniku/ci @fermatslibrary
Please elaborate .
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Kerthorok 29. sij
Odgovor korisniku/ci @Yps057 @fermatslibrary
For any polynomial of degree 4 or less, there is a formula to extract the roots in terms of radicals of the coefficients of the polynomial (e.g. the Quadratic Formula). However, for a general polynomial of degree 5 or higher, there is no such formula.
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Just a guy 29. sij
Odgovor korisniku/ci @fermatslibrary
Wow. The complexity rises quickly. The solution to ax^3+bx^2+cx+d=0 is in the 1st picture (which I've never seen). One expects three solutions (here's one), Two solutions might be imaginary (2nd picture).
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FF 29. sij
Odgovor korisniku/ci @Jack_Forte1 @fermatslibrary
Interestingly, in formulating and proving this equation, Rafael Bombelli, coined the value of "i" as a constant, in order to solve the inconsistency. Both "i"s would cancel itself, giving a numerical answer for x. He thought it was a pure coincidence and ignored his discovery.
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Guy Larange 29. sij
Odgovor korisniku/ci @fermatslibrary
And died in a duel next day!?
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dockast 29. sij
Odgovor korisniku/ci @glarange72 @fermatslibrary
No, that was Galois, not Abel! Abel was murdered by Kain... 😉
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Levi 🧪 29. sij
Odgovor korisniku/ci @fermatslibrary
There also happens to be a non-algebraic closed formula for the roots of the quintic which uses modular forms and elliptic integrals:
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Merly 29. sij
Odgovor korisniku/ci @fermatslibrary
So it is possible to solve up to degree 4?
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. 29. sij
Odgovor korisniku/ci @Pepoluco @fermatslibrary
Yes u can solve upto degree 4
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Kerthorok 29. sij
Odgovor korisniku/ci @fermatslibrary
There seems to be some confusion in the comments. This theorem does not imply that ALL quintics are unsolvable. For example (×-1)^5=0 is trivial to solve. It simply states that given an arbitrary a,b,c,d,e there is no solution in terms of radicals of those coefficients.
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Kerthorok 29. sij
Odgovor korisniku/ci @fermatslibrary
Thus this theorem really states that there are SOME quintics that can't be solved by radicals, like x^5-x-1 for example. Galois Theory specifies when polynomials are solvable by identifying their Galois Groups and determining if those groups are solvable.
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