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Keywords:
Galois theory; finite group; permutation; radical
Summary:
We illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed by radicals, i.e., by the operations $+,\,-,\,\cdot,\,:\,$, and $\root n \of{\cdot}$. Therefore, higher order polynomial equations are usually solved by approximate methods. They can also be solved algebraically by means of ultraradicals.

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