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On Euler’s proof of the fundamental theorem of algebra. (English) Zbl 1235.12004

It is well known that Euler’s proof of the Fundamental Theorem of Algebra had gaps. In this article, the authors use Euler’s ideas for studying the surjectivity of polynomial mappings and show that Euler’s proof can be completed using modern algebra.
Euler’s formulation of the Fundamental Theorem is the following: every polynomial \(f \in \mathbb R[x]\) with degree \(\geq 2\) has a real factor of degree \(2\). This in turn is a consequence of the fact that real polynomials of \(2\)-power degree \(2k \geq 4\) always split into two factors of degree \(k\). This means that the map \(\mathbb R^k \times \mathbb R^k \longrightarrow \mathbb R^{2k}\) induced by the multiplication of monic polynomials of degree \(k\) is surjective. This problem is then translated into algebraic geometry and used as a motivation for studying the problem of finding criteria that guarantee that a finite \(K\)-algebra has a \(K\)-rational point.
This problem is solved by introducing the mapping degree \(\delta(T)\) of polynomial mappings \(T: K^N \longrightarrow K^N\) over real closed fields \(K\) (this is an algebraic version of the mapping degree from differentiable topology). The main result (Thm. 4.6) on the mapping degree is that \(T\) is surjective if \(\delta(T) \neq 0\). The actual calculation of the mapping degree of the Vieta mapping \(V_{(k,\ell)}: K^k \times K^\ell \longrightarrow K^{k+\ell}\) then implies the Fundamental Theorem of Algebra: Every polynomial \(f \in K[X]\) of degree \(\geq 2\) over a real closed field \(K\) has a quadratic factor in \(K[X]\).

MSC:

12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
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