Background Information

The proof of Abel's Theorem showed that for every n ≥ 5 there exists a polynomial ƒ of degree n over Q such that at least one root of ƒ cannot be expressed in radicals. A natural question is whether a stronger result may actually be true, namely: For each value of n ≥ 5, does there exist a polynomial ƒ of degree n over Q such that none of the roots of ƒ can be expressed in radicals? As was pointed out just after the proof of Abel's Theorem, the quintic polynomial used there actually has this stronger property. It follows from Problems 5, 6, and 7 of the same section that for all prime values of n > 5, there also exists such a polynomial. In a slightly different vein, if we drop the restriction that the coefficient field be Q, Problem 8 of that section shows that for every n ≥ 5 there always exists some polynomial of degree n, none of whose roots can be obtained by a sequence of radical extensions of the coefficient field.

All of these partial results were obtained by finding polynomials whose Galois groups were Sn, for various values of n. Thus it is natural to attack the previous question by asking: For what values of n does there exist a polynomial over Q whose Galois group is Sn?