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Galois Theory, the theory of polynomial equations and their solutions, is one of the most fascinating and beautiful subjects of pure mathematics. Using group theory and field theory, it provides a complete answer to the problem of the solubility of polynomial equations by radicals: that is, determining when and how a polynomial equation can be solved by repeatedly extracting roots using elementary algebraic operations. This textbook contains a fully detailed account of Galois Theory and the algebra that it needs and is suitable both for those following a course of lectures and the independent reader (who is assumed to have no previous knowledge of Galois Theory). The second edition has been significantly revised and re-ordered; the first part develops the basic algebra that is needed, and the second a comprehensive account of Galois Theory. There are applications to ruler-and- compass constructions, and to the solution of classical mathematical problems of ancient times. There are new exercises throughout, and carefully-selected examples will help the reader develop a clear understanding of the mathematical theory.
It is likely that the reader has already met the concept of a group. It was Galois who first understood the imporance of groups in the study of the roots of a polynomial equation; since then, group theory has blossomed, and developed as a subject in its own right. In this chapter we simply develop those parts of the theory which we shall need later; one of the main purposes is to explain the notation and terminology that we shall use.
The reader will probably be familiar with the notion of a real or complex vector space. In Galois theory we need to consider vector spaces over an arbitrary field 𝐾: we replace the field of scalars ℝ or ℂ by an arbitrary field 𝐾
There are many field extensions. How do we recognize when an extension 𝐿: 𝐾 is a splitting field extension? For this we need the notion of a normal extension.
Normality is a property that an extension may or may not have. Separability is different; most extensions of interest are separable, and we shall have to work hard to find examples of non-separable extensions.
Although we shall obtain further general results in Galois theory in the next chapter, Galois theory’s first intent is to throw light on the solution of polynomial equations. We now pause to see how the theory that we have developed so far relates to the solution of equations of low degree. In this chapter, we consider the solution of monic quadratic, cubic and quartic polynomial equations over a field 𝐾.
We have seen that in order to deal with cubic polynomials it is helpful to have cube roots of unity at our disposal. In this chapter and the next we shall consider splitting fields and Galois groups of polynomials of the form 𝑥𝑚 − 1 and 𝑥𝑚 − θ over a field 𝐾.
One of the main topics of Galois theory is the study of polynomial equations. In order to consider how we should proceed, let us first consider some rather trivial and familiar examples.
The results of the two preceding chapters, together with the fundamental theorem of Galois theory, suggest that, provided that we can construct enough roots of unity, a separable polynomial is solvable by radicals if and only if its Galois group can be built up in some way from abelian groups. We shall see that this is indeed so.
What about quintic polynomials? There are quintic polynomials in ℤ[𝑥] which cannot be solved by adjoining radicals. This was proved by Abel in 1824, some five years before the fundamental work of Galois.
Recall that a ring homomorphism from a field 𝐿1 to a field 𝐿2 is a monomorphism, and that if 𝐿 : 𝐾 is an algebraic field extension and that τ : 𝐿 → 𝐿 is a homomorphism which is fixed on 𝐾, then τ is an automorphism of 𝐿 (Theorem 4.9).
Many of the problems that exercised Greek mathematicians and their successors were geometric, and in particular concerned constructions using ruler (straight edge) and compasses. Here are the three most important.
In this chapter, we consider a field 𝐹 with a finite number 𝑞 of elements, and its Galois theory. Its prime subfield is isomorphic to ℤ𝑝 for some prime number 𝑝, and we identify it with ℤ𝑝. 𝐹 is a finish-dimensional vector space over ℤ𝑝, of dimension 𝑛, say, so that 𝑞 = 𝑝𝑛.
If 𝑓 ∈ ℚ[𝑥] we can consider 𝑓 as an element of ℂ[𝑥], and then 𝑓 splits over ℂ. We therefore have the comforting conclusion that, whenever 𝑓 ∈ ℚ[𝑥], we can find a splitting field extension for 𝑓 which is a subfield of the fixed field ℂ.
In this chapter, we consider which regular polygons can be constructed by ruler and compasses. This requires more than the material of . Indeed, the chapter can be seen as an extended exercise of the theory ofto , together with the theory of 𝑝-groups and soluble groups. We need one more ingredient.
Suppose first that 𝑓 ∈ ℚ[𝑥]. As we have seen, we may factorize 𝑓 in an essentially unique way into irreducible factors. Further, 𝑓 can be written as λg, where λ ∈ ℚ and 𝑔 is a primitive element of ℤ[𝑥].