What is Galois Theory Anyway?

The Basic Idea

Perhaps you've heard of Évariste Galois? (Pronounced "GAL-wah.") You know, the French mathematician who died tragically in 1832 in a duel at the tender age of 20? (Supposedly over a girl! C'est romantique, n'est-ce pas?) Well, today we're taking a bird's-eye view of his most well-known contribution to mathematics: the appropriately named Galois theory. The goal of this post is twofold:

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If you are a student about to study Galois theory, I hope the info below will serve as a small appetizer to your main course. In the "From English to Math" section below, we'll take a brief survey of the ideas that appear in a standard graduate course so that when you start doing exercises, you at least have a bird's-eye-view of what's going on.

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Even if you're not about to study Galois theory and are just curious, this post is also for you! The info here should be accessible to anyone with at least an undergrad background in abstract algebra. I'm going to leave out a lot of technical details (but you don't that mind, do you?) as my goal here is just to convey some main ideas. I hope this may whet your appetite to study further.

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In a word, Galois Theory uncovers a relationship between the structure of groups and the structure of fields. It then uses this relationship to describe how the roots of a polynomial relate to one another.

More specifically, we start with a polynomial $f(x)$. Its roots live in a field (called the splitting field of $f(x)$). These roots display a symmetry which is seen by letting a certain group (called the Galois group of $f(x)$) act on them. And we can gather information about the group's structure from the field's structure and vice versa via the Fundamental Theorem of Galois:

Now why would anyone care about permuting the roots of a polynomial? What good does that do? Well, the quadratic formula was well-known by the time Galois came along. (It actually dates back to the Babylonians). So naturally, mathematicians wondered, "Does an analogous formula exist for polynomials of higher degree?" In other words, can the roots of an nth degree polynomial be written down as some algebraic combination $(+,-,\times,\div,\sqrt)$ of the polynomial's coefficients? It turns out the answer is "Yes!" when $n\leq 4$, but "No," for any $n\geq 5$.

It was precisely Galois' study of permutation groups of the roots of polynomials that led to his discovery of a necessary and sufficient condition for finding a such a formula.* The condition (which eluded mathematicians for over 300 years!) becomes elegantly clear when the problem is translated from the language of field theory to that of group theory. Galois theory is the dictionary which makes this possible.

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From English to Math

The Field Story

Suppose $F$ is a field. Then the polynomial ring $F[x]$ is a (Euclidean domain and hence a) unique factorization domain. This means any polynomial $f(x)\in F[x]$ can be factored uniquely as a product of irreducible polynomials. Now we know that any root of $f$ must be a root of one of those irreducible factors, but we may not know how and where to find those roots. It turns out that passing to field larger than $F$ helps produce those roots.

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Example: Consider the irreducible polynomial $f(x)=x^2+1$ with coefficients in $\mathbb{Q}$. One of its roots is $i=\sqrt{-1}$. Even though $i

ot\in\mathbb{Q}$, we can find a bigger field which contains it, namely $\mathbb{Q}(i)=\{a+bi:a,b\in\mathbb{Q}\}$. ($\mathbb{Q}(i)$ is "bigger" since $\mathbb{Q}\subset\mathbb{Q}(i)$.)

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In the example above, we say $\mathbb{Q}(i)$ is an extension field of $\mathbb{Q}$. And in general, any field $K$ which contains a smaller field $F$ is called an extension of $F$, and $F$ is referred to as the base (or ground) field. But so far we've only talked about a single root of our polynomial $f(x)$. What about all of its roots? Can we find an extension of $F$ which contains all of the roots of a general $f(x)$? Again the answer is yes, and that field is called the splitting field of $f(x)$. (Technically, the splitting field the smallest extension of $F$ which contains all the roots of $f(x)$.)

(Aside: We could also turn this question on its head. Suppose we have a field $F$ and an extension $K$, and we pick a random element** $\alpha\in K$. Can we find a polynomial in $F[x]$ which has $\alpha$ as a root? This time, the answer is not always yes! But in the cases when it is, we say $\alpha$ is an algebraic element. Moreover, if the answer is yes for every element of $K$, we say $K$ is an algebraic field. As we've discussed before, algebraic elements are sort of like limit points in topology/analysis.)

Now the crux of The Field Story is the construction of such splitting fields. This construction is analogous to building a tower from the ground up - one floor at at time. We begin with the ground field $F$, and one by one adjoin to $F$ the roots of $f(x)$ until we obtain the field $K$ (the splitting field) which contains all the roots of $f$. I'm doing a lot of hand-waving here, but we eventually obtain a tower of fields which looks something like this:

where $F_{i+1}$ is bigger than $F_i$ because it contains (at least) one more root of $f(x)$. It's the structure of this tower of fields which is mirrored in the structure of the Galois group associated with $f(x)$. And that group is our next topic of discussion.

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The Group Story

Do you remember those verb commercials from the early to mid 2000s? You know, the ones full of hyper-happy, active kids encouraging you to do some physical activity? (No? Just Google it, you'll see.) For some reason I do. The tag-line was

VERB: It's what you do.

And in the context of mathematics, groups are very much like verbs! They do stuff. In particular, a group acts on a set by shuffling elements around. We can gather a lot of information about the group and about the set it acts on via this action. So if you ever want to create a TV commercial about mathematical groups,

GROUPS: It's what they do.

would be a telling tag-line. But how does this relate to Galois theory? Well the Galois group $G$ associated to an $n$th degree polynomial $f(x)\in F[x]$ has its own action. In particular, the elements of $G$ are automorphisms $\sigma:K\to K$ where $K$ is the splitting field of $f$. (Recall a field automorphism is just an isomorphism from a field to itself.) This group is isomorphic to (i.e. has the same structure as) a subgroup, called a permutation group, of the symmetric group $S_n$.

Since the raison d'être of a permutation group is to permute things, it's not too hard to believe that $G$ acts on the splitting field $K$ by permuting the roots of $f(x)$. So suppose for a moment that $f(x)\in F[x]$ can be written as a product of irreducible factors $f(x)=f_1(x)f_2(x)\cdots f_k(x)$. Then $G$ will permute the roots of $f_1$ among themselves, and the roots of $f_2$ among themselves, and so on. In other words, $G$ is said to act transitively on the irreducible factors of $f$. For instance, in the drawing below, $G$ is some subgroup of $S_n$ (for the sake my illustration, suppose*** $n\geq 9$) which contains a product of two 2-cycles, $(13)(24)$, one three cycle, $(567)$, and a transposition $(n \; n-1)$.

In general, once we know the Galois group of $f$, we can analyze its subgroup structure. Galois' insight was to notice that if the structure of $G$ was such that it has a chain of subgroups

(where $e$ is the identity) such that $H_i\triangleleft H_{i+1}$ is normal and $H_{i+1}/H_{i}$ is abelian, then and only then can we write down an explicit algebraic expression (like the quadratic formula) for the roots of $f(x)$. In this case, we say the group $G$ is solvable. By the Fundamental Theorem of Galois, the ability to write down this chain of subgroups corresponds to the ability to write down a particular tower of subfields of the splitting field $K$ of $f$. So the solvability of our polynomial $f$ amounts to knowing something about the structure of its splitting field, or equivalently the structure of its Galois group. The fact that there is no algebraic formula for the roots of polynomials of degree 5 and higher is due to the fact that the symmetric group $S_n$ for $n\geq 5$ is not solvable!

And with that, we end our bird's-eye view of a course in Galois Theorey. Whew! (Kudos if you made it through the whole thing!) I hope you've gotten a little taste of what Galois Theory is about. But this is really just the tip of the iceberg. For further reading, I highly recommend Basic Abstract Algebra (2 ed., ch. 15-17) by Bhattacharya, Jain, and Nagpaul. This book is very approachable at the undergraduate level. And of course, Abstract Algebra (3 ed., ch. 13-14) by Dummit and Foote is a classic for both undergraduate and graduate students.

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Footnotes

* In fact, Galois himself invented the concept of a group for this very purpose!

**Edited May 21, 2016: This sentence originally read, "...a random element $\alpha\in F$" and did not mention the extension field $K$ (a rather boring case)! Many thanks to a reader for pointing this out.

*** In a typical course, you'll usually play with polynomials of degree two, three, four, and five since the group structures of $S_2$, $S_3$, $S_4$, and $S_5$ are more manageable than that of $S_n$ for $n\geq 6$.