My goal in this text is to develop Galois theory in as accessible a manner as possible for an undergraduate audience.

Consequently, algebraic numbers and their minimal polynomials, objects as concrete as any in field theory, are the central concepts throughout most of the presentation. Moreover, the choices of theorems, their proofs, and (where possible) their order were determined by asking natural questions about algebraic numbers and the field extensions they generate, rather than by asking how Galois theory might be presented with utmost efficiency. Some results are deliberately proved in a less general context than is possible so that readers have ample opportunities to engage the material with exercises. In order that the development of the theory does not rely too much on the mathematical expertise of the reader, hints or proof sketches are provided for a variety of problems.

The text assumes that readers will have followed a first course in abstract algebra, having learned basic results about groups and rings from one of several standard undergraduate texts. Readers do not, however, need to know many results about fields. After some preliminaries in the first chapter, giving readers a common foundation for approaching the subject, the exposition moves slowly and directly toward the Galois theory of finite extensions of the rational numbers. The focus on the early chapters, in particular, is on building intuition about algebraic numbers and algebraic field extensions.

All of us build intuition by experimenting with concrete examples, and the text incorporates, in both examples and exercises, technological tools enabling a sustained exploration of algebraic numbers.

The Subgroups of S4

There are eleven conjugacy classes of the 30 subgroups of S4, as follows:

1. (1) {〈0〉}, the class consisting of the single subgroup of order 1.

2. (2) The class of all subgroups of order 2 that are generated by a single transposition, that is, subgroups of the form 〈(ij)〉 for distinct i and j. This class has six elements.

3. (3) The class of all subgroups of order 2 that are generated by the product of two disjoint transpositions, that is, subgroups of the form 〈(ij)(kl)〉 for distinct i, j, k, and l. This class has three elements.

4. (4) The class of all subgroups of order 3. These are generated by a 3-cycle and take the form 〈(ijk)〉 for distinct i, j, and k. This class has four elements.

5. (5) The class of all cyclic subgroups of order 4. These are generated by a 4-cycle and take the form 〈(ijkl)〉 for distinct i, j, k, and l. This class has three elements.

6. (6) The class of all subgroups of order 4 isomorphic to the Klein 4-group ℤ/2ℤ ⊕ ℤ/2ℤ that are generated by two transpositions. These subgroups have the form 〈(ij), (kl)〉 for distinct i, j, k, and l. This class has three elements.

7. (7) {〈(12)(34), (13)(24)〉}, the class consisting of the unique subgroup of order 4 isomorphic to the Klein 4-group ℤ/2ℤ ⊕ ℤ/2ℤ that is generated by two products of two transpositions.

8. (8) The class of all subgroups of order 6. These have the form 〈(ijk), (ij)〉 for distinct i, j, and k. This class has four elements.

9. […]

Knowing the Galois correspondence for subfields of ℂ and some strategies for the determination of Galois groups of field extensions is, of course, just the beginning. It is of interest to determine the Galois groups of families of polynomials, to determine field extensions of a ℚ with a specified Galois group, to find generalizations of the Galois theory of subfields of ℂ to arbitrary fields, and to apply the tools of Galois theory to solve problems further a field. In this chapter, we give introductions to several of these topics.

Roots of Unity and Cyclotomic Extensions

Perhaps the most basic polynomials are those of the form Xn − 1. In this section, we study their roots, called roots of unity, and the field extensions they generate. We first define roots of unity and cyclotomic extensions.

Definition 30.1 (Root of Unity, Primitive Root of Unity). A root of unity ω is a root, in the complex numbers, of a polynomial of the form Xn − 1 for some n ∈ ℕ. We say that ω is a primitive root of unity of order n, or a primitive nth root, if ω is a root of Xn − 1, but not of Xm − 1 for any 1 ≤ m < n.

Note that the definition of a root of unity ω is equivalent to saying that ω is an element of finite order in the multiplicative group ℂ* of ℂ, and it will be useful to view roots of unity in this way. You may wish to recall some results on finite cyclic groups, particularly concerning which elements of a cyclic group are generators of that group.

Over any field K ⊂ ℂ, we have associated to each algebraic number α a polynomial mα,K and a field K(α). In this chapter,we consider to what extent these objects are unique, that is, whether a given polynomial or field might be generated by other algebraic numbers β as well. In the first section, we determine all of the algebraic numbers to which a minimal polynomial is associated. In the second, we find that if a field is generated by an algebraic number, it is generated by each of an infinite number of algebraic numbers, and we introduce a notion of size of a generated field. Finally, in section 14 we introduce tools for exploring fields generated by an algebraic number.

Minimal Polynomials Are Associated to Which Algebraic Numbers?

We seek to describe the relationship between algebraic numbers and minimal polynomials as precisely as possible, and we begin by observing how many roots a polynomial may have in a field. We will then relate these roots to linear factors of the polynomial over the complex numbers, and we finally consider what we may say about minimal – that is, irreducible – polynomials.

A Polynomial of Degree n Has at Most n Roots in Any Field Extension

We first present a theorem over arbitrary fields that bounds the number of roots a polynomial may have in a field.

Theorem 11.1. Let K be a field, p ∈ K[X] a nonzero polynomial, n = deg(p), and L a field containing K. Then p has no more than n roots in L.

We now extend our investigation to fields generated over a field K by more than one algebraic number, or multiply generated fields. We saw a hint of such fields in section 12.3, where we considered a field K(α) generated by an algebraic number α over a field K(β), itself generated over K by an algebraic number β. In this chapter, we consider several questions concerning fields generated over K by more than one algebraic number:

• whether such a field may be generated over K by a single algebraic number;

• whether such a field is of finite dimension over K; and

• how to specify the structure of such fields via an isomorphism to a quotient ring of a polynomial ring, as we did in section 8.3.

We also examine an important class of fields generated by several algebraic numbers: those that are generated by all of the roots of a polynomial. We then study how to determine isomorphisms from one multiply generated field to another, particularly when the fields are splitting fields. At the end of the chapter, we consider the results of this chapter in the general field-theoretic setting of simple, finite, and algebraic extensions.

Fields Generated by Several Algebraic Numbers

Definition 16.1. Let K be a subfield of ℂ, and let α and β be two algebraic numbers. We denote by K(α, β) the identical fields K(α)(β) = K(β)(α), which we call the field generated by α and β over K.

In support of the preceding definition, note that we may view the set S of all arithmetic combinations of α, β, and elements of K in several ways.

We are now ready to introduce one of the most elegant results in algebra, the Galois correspondence. This correspondence gives us a framework for understanding the relationships between the structure of a splitting field over a field K and the structure of its group of automorphisms over K. Once we establish this correspondence, we go on to study in some detail how the group provides some distinguishing characteristics of the various conjugates over K of an element of the splitting field.

Normal Field Extensions and Splitting Fields

The property of splitting fields that encapsulates much of the information necessary for proving steps in the Galois correspondence is that of normality. In this section, we introduce the property and explore its connection with splitting fields.

Definition 23.1 (Normal Field Extension in ℂ). Let K be a subfield of ℂ. An algebraic field extension L/K is normal if every polynomial f ∈ K[X] that is irreducible over K and that has at least one root in L contains n = deg(f) roots in L.

Note that since the roots of an irreducible polynomial in K[X] are distinct (Theorem11.3), this definition is equivalent to

Definition 23.2 (Normal Field Extension). An algebraic field extension L/K is normal if every polynomial p ∈ K[X] that is irreducible over K and that has at least one root in L factors into linear terms over L.

Clearly a splitting field of an irreducible polynomial p ∈ K[X] satisfies the condition for the particular polynomial p, but the interest of the property lies in whether the condition is satisfied for all irreducible polynomials q ∈ K[X]: each such polynomial must have either none of its roots in L or all of its roots in L.