Conditions for the insolvability of the quintic equation \(x^5+ax+b=0\) (Q2721574)
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scientific article; zbMATH DE number 1616248
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Conditions for the insolvability of the quintic equation \(x^5+ax+b=0\) |
scientific article; zbMATH DE number 1616248 |
Statements
2 January 2002
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quintic equation
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solvability by radicals
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Conditions for the insolvability of the quintic equation \(x^5+ax+b=0\) (English)
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The authors consider polynomials of the form \(f(X)=X^5+aX+b\) with \(a,b\) being rational integers and give certain conditions assuring the insolvability of the equation \(f(X)=0\) by radicals. In particular they show that this happens in the following cases: (a) \(a\) is odd and \(b\equiv 2\pmod 4\), (b) \(b\) divides \(a\), (c) there exists a prime \(p\equiv 3\pmod 4\) with \(p\parallel a\), \(p^2|b\).
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