Divisible By Pi

So, I’ve already touched on rings and modules. That only leave fields and Galois Theory to touch on in this series on university algebra (and my university subjects in general).

When I first introduced fields during my discussion of different types of rings, I defined a field as ‘a commutative division ring‘. While this is certainly a true and useful definition, it is not the most intuitive definition of a field and it gives the false impression that the study of fields grew out of the study of the properties of rings — in fact field theory predates ring theory, having been developed in the early nineteenth century by Niels Abel and Évariste Galois (for whom abelian groups and Galois theory are named, respectively), while ring theory owes its existence more to Dedekind and Hilbert, who came along later that century.

Intuitively, fields are extremely easy things to understand: Fields are sets that behave in the way we normally expect numbers to behave — that is to say, under the operations of addition, subtraction, multiplication and division a field gives us sensible answers and doesn’t prohibit us from performing any of those operations as expected (with the exception of division by zero, which is prohibited for extremely good reasons). The rational numbers, the real numbers and the complex numbers are ‘familiar’ (in some sense) examples of fields — less obviously, the integers modulo seven (or indeed modulo any prime number) are an example of a field of finite size.

While the benefit of studying fields may not be immediately apparent, the payoff comes in part by studying the properties of finite fields and how they relate to polynomials. Taking the rational numbers as a fundamental field we may be interested in studying, for example, we are often interested in examining a particular polynomial and extending the rational numbers just enough to create field that contains all of the roots of that polynomial. For example, take the polynomial

.

The roots of this polynomial are  — however these do not lie in the field of rational numbers, . They do lie in the field of real numbers, , but at the same time so do many other numbers that we are not interested in.

There is a field that is ‘just right’ for this polynomial — a Goldilocks field, as it were — that extends the field of rational numbers just far enough to include the roots of this polynomial without including any other roots. This field is obtained by taking all the numbers that can be written as the sum of a rational number with a rational multiple of the square root of two, that is –

– and we call this field . Fields such as this (i.e. ‘smallest possible’ fields) are appropriately named splitting fields.

An extremely powerful technique for analysing polynomial equations and determining relationships between the roots of particular equations is Galois Theory.

Évariste Galois was in some sense the early mathematical equivalent of a rock star — his collected works total only at around sixty pages, yet within those sixty pages he became one of the pioneers of group theory and founded the principles upon which a great deal of powerful maths has been built. A politically charged and rebellious man, he died at age twenty after having been shot in the abdomen during a duel —  generations of mathematicians have felt uneasily inadequate ever since.

Galois Theory itself is an ingenious coupling of field theory and group theory. First, we define the Galois group: For a field extension , the Galois group is the set of all F automorphisms in E — that is, the set of all isomorphisms (remember them from modules?) from E to itself that leave every element of F fixed.

This extremely abstract concept allows us to reduce problems from field theory relating to polynomials to problems in the comparatively simpler domain of group theory — it does this by way of a theorem called the Fundamental theorem of Galois theory (a name only slightly less foreboding than those we encountered in our discussion of modules). An explicit statement of this theorem would likely be a little bit overwhelming — given, after all, that I had a semester to prepare for it while you have only had two blog posts — so we’ll delay that for a later more adventurous day. Essentially, the theorem demonstrates that there is a correspondence between the subfields of the splitting field of a polynomial, and the subgroups of that splitting field’s Galois group over the rationals.

The polynomial we dealt with up above is, sadly, rather a boring case so far as splitting fields and Galois groups go — the only subfields of the splitting field turn out to be itself, and the base field . (Incidentally, the corresponding subgroups of the Galois group are the trivial group consisting of a single element and the entire Galois group, respectively.)

Instead, we’ll examine a slightly more interesting polynomial:

The splitting field for this polynomial is, unsurprisingly, ; the Galois group over the rational numbers consists of four functions:

1. The identity function, which leaves all elements unchanged.
2. The function which swaps  and , but leaves  untouched.
3. The function which swaps  and , but leaves  untouched.
4. The function which swaps  and , and also swaps  and .

In addition to the splitting field and Galois group, we now also have proper subfields and subgroups. These subfields of can be matched with unique corresponding subgroups, determined by which automorphisms leave the subfield in question unchanged. These are given by:

• (the entire splitting field), corresponding to the trivial subgroup.
• , corresponding to the subgroup containing functions 1 and 2.
• , corresponding to the subgroup containing functions 1 and 3.
• , corresponding to the subgroup containing functions 1 and 4.
• (just the rational numbers), corresponding to the entire Galois group.

These techniques and the ability to find correspondences between fields and groups are more than mere mathematically curiosities, however. With only a little bit of extension work and a slightly deeper understanding of the concepts of group theory we can use the tools of Galois theory to answer rather deep mathematical questions — for example, Galois theory allows us to prove that there is no general expression for the roots of a quintic equation (degree five polynomial) analogous to the quadratic formula, which tells us the roots of the polynomial  are given by

or Cardano’s formula, which gives us a general expression for the roots of a cubic equation (degree three polynomial). Sadly, such a proof is definitely beyond the scope of a single blog post.

I am under no illusion that for many people the contents of these last three posts on algebra have likely been more difficult to comprehend than the posts that preceded them — this is in part due to the abstract nature of the subject, and in part because of my desire to go into a little bit more depth about what exactly is involved in studying pure mathematics. To those people who may have struggled through these posts and feel somewhat lost but are still interested I offer this reassurance — you are not alone. The topics I have touched upon tend to require going over even for maths students in order to get a feel for things.

I’d implore anybody who found these posts interesting to do the following: Bookmark these pages, give yourself a rest for a few days, and then come back and read them again. You still might not understand all of it, but I’d wager that you will at least feel more familiar with the topic and hopefully will be able to take something new away with you. I’ve aimed for both brevity and clarity in my explanations — my success will primarily be a matter of how well I convey at least the feeling of the subject matter to you — and in that regard I have tried to emulate the mathematical giant David Hilbert, who I will leave with the final word on this subject:

An old French mathematician said: A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street. This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us.

— David Hilbert

Filed under: Maths, University