Let the polynomial $$P(x) := x^n+a_1x^{n-1}+\ldots+a_n$$ with complex coefficients $a_j$ be irreducible, i.e. irreducible in the field $\mathbb{Q}(a_1,\,\ldots,\,a_n)$ , of its coefficients. If the equation $P(x) = 0$ , can be solved algebraically and if all of its roots are real, then no root may be expressed with the numbers $a_j$ using mere real radicals unless the degree $n$ of the equation is an integer power of 2.

Bibliography

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K. V¨AISÄLÄ: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No. 17.    Kustannusosakeyhtiö Otava, Helsinki (1950).