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There are various ways in which to represent the elements of a finite field. Since every finite field is simple, it has the form = ()( for some ∈ and so the elements of are polynomials in of degree less than deg (). Another way to represent the elements of a finite field is to use the fact that ()* is cyclic, and so its elements are all powers of a group primitive element.

Polynomials of the form − , where 0 ≠ ∈ , are known as . Even though binomials have a simple form, their study is quite involved, as is evidenced by the fact that the Galois group of a binomial is often nonabelian. As we will see, an understanding of the binomial − 1 is key to an understanding of all binomials.

Continuing our discussion of binomials begun in the previous chapter, we will show that if is a splitting field for the binomial − , then = () where is a primitive th root of unity. In the tower the first step is a cyclotomic extension, which, as we have seen, is abelian and may be cyclic. In this chapter, we will see that the second step is cyclic of degree. | and can be chosen so that min(()) = − . Nevertheless, as we will see in the next chapter, the Galois group ( need not even be abelian.

We now turn to the question of when an arbitrary polynomial equation () = 0 is . Loosely speaking, this means (for char() = 0) that we can reach the roots of () by a finite process of adjoining th roots of existing elements, that is, by a finite process of passing from a field to a field (), where is a root of a binomial − , with ∈ . We begin with some basic facts about solvable groups.

We continue our study of binomials by determining conditions that characterize irreducibility and describing the Galois group of a − binomial in terms of 2 × 2 matrices over ℤ. We then consider an application of binomials to determining the irrationality of linear combinations of radicals. Specifically, we prove that if ,...., are distinct prime numbers, then the degree of over ℚ is as large as possible, namely, . This implies that the set of all products of the form where 0 ≤ () ≤ − 1, is linearly independent over ℚ For instance, the numbers are of this form, where = 2, = 3. Hence, any expression of the form where ∈ ℚ, must be irrational, unless = 0 for all .

In this chapter, we look briefly at families of binomials and their splitting fields and Galois groups. We have seen that when the base field contains a primitive th root of unity, cyclic extensions of degree | correspond to splitting fields of a single binomial − . More generally, we will see that abelian extensions correspond to splitting fields of families of binomials. We will also address the issue of when two families of binomials have the same splitting field.