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Recenly I've been wondering whether it's possible to prove the FTA for all polynomials of degree $n \le 4$ without utilizing advanced maths but, at most, basic linear algebra (concepts such as eigenvectors, eigenvalues, determinants etc.). I've tried to give this some thought but I've been only able to make very naïve and futile statements such as "all real numbers can be represented as complex numbers " and "some quadratics and quartics, such as $p(x)=x^2 + 1$ and $p(x)=x^4 -1$, contain complex solutions". Now, I would be really greatful, if it is indeed possible to prove the FTA for all polynomials with degree $n \le 4$, if someone could provide a reference of some sort.

Eric Wofsey
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user301644
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    Pulling out the quadratic/cubic/quartic formula and showing that they are correct is a proof of this. The only sticking point is choosing the right branch of the radicals. – Milo Brandt Dec 30 '15 at 01:48
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    I vaguely remember doing this for cubics, back in 2010 when I was first discovering that, hey, if you think about things for awhile, hey presto, sometimes you work things out! Can't remember the details, but in retrospect the argument was probably circular, because it probably used facts about matrices with complex entries that relied on FTA to prove in the first place. – goblin GONE Dec 30 '15 at 01:53
  • The Wikipedia has some nice proofs: https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra – Simply Beautiful Art Dec 30 '15 at 02:15
  • The most elegant way is to use Liouville's Theorem. You may start with the definition of holomorphic functions in a complex variable book. An example is this one: http://www.math.s.chiba-u.ac.jp/~yasuda/ippansug/CV-bookfi.pdf – GNUSupporter 8964民主女神 地下教會 Dec 30 '15 at 09:40

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Let $p\in\mathbb{R}[X]\setminus\mathbb{R}$ be a polynomial of degree at most $4$.

  • If $\deg(p)=1$, then you can easily explicit the root of $p$.

  • If $\deg(p)=2$, same remark.

  • If $\deg(p)=3$, you can also explicit the roots of $p$ using Cardano's method.

  • If $\deg(p)=4$, you can also explicit the roots of $p$ using this time Ferrari's method.

Note that if $\deg(p)\geqslant 5$, in the general case, you won't be able to find such formulas for the roots of $p$, it's the Ruffini-Abel theorem

C. Falcon
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