Galois theory is an advanced and abstract branch of mathematics that often comes up in interviews for research positions or graduate studies in mathematics Mastering the key concepts and being able to eloquently explain them shows mathematical maturity and analytical skills While Galois theory questions won’t make or break your chances if you’re just starting out, they are a rite of passage for aspiring mathematicians.
In this article, I’ll go over some of the most common Galois theory interview questions, provide example answers, and share tips on how to prepare.
Background on Galois Theory
First, a quick refresher on what Galois theory is all about. Developed by French mathematician Évariste Galois in the 1830s it provides a connection between field theory and group theory. Galois theory studies algebraic equations and their solutions focusing on which equations can be solved by radicals.
Some key concepts include
- Field extensions – enlarging a field by adding elements like square roots or cube roots
- Splitting fields – smallest field extension where a polynomial splits into linear factors
- Galois groups – groups capturing the symmetry of roots of polynomials
- Solvability – whether or not a polynomial equation can be solved by radicals
Galois realized that if a polynomial’s Galois group is solvable, then you can solve the equation by radicals. This revealed why the general quintic equation couldn’t be solved radicals – its Galois group, S5, is not solvable for n≥5.
Sample Galois Theory Interview Questions
Here are some common technical questions on Galois theory that come up in interviews:
Q1: Explain the Fundamental Theorem of Galois Theory.
The Fundamental Theorem of Galois Theory establishes a correspondence between subfields of a Galois field extension and subgroups of its Galois group.
Specifically, it states:
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For any subfield F of a Galois extension E, there is a corresponding subgroup H of the Galois group G such that F consists of all elements in E fixed by H.
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Conversely, for any subgroup H of G, the set of elements in E fixed by H form a subfield of E.
Q2: What is the difference between solvable and soluble groups? Give examples of each.
This is a terminology question testing that you know the precise definitions:
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A solvable group is one that has a subnormal series where each factor group is abelian. Example: the symmetric group S3.
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A soluble group is one that has a subnormal series where each factor group is cyclic of prime order. Example: the alternating group A4.
Soluble implies solvable, but not vice versa. S3 is solvable but not soluble, while A4 is both solvable and soluble. Knowing the distinction shows you understand group theory subtleties.
Q3: Prove that S5 is not a solvable group for n≥5.
This asks you to show directly why the symmetric group S5 is not solvable, implying the quintic is unsolvable by radicals. Here is a sketch of the proof:
- Suppose S5 has a subnormal series 1 ⊲ A ⊲ B ⊲ S5.
- The factor groups must be abelian, so A ⊲ S5 implies A is abelian.
- But A must contain an element of order 5, implying A is all of S5, a contradiction.
- Therefore, no such subnormal series exists, so S5 is not solvable for n≥5.
This demonstrates you understand the group theory argument for the quintic’s unsolvability.
Q4: If a polynomial has non-abelian Galois group G, prove it is unsolvable by radicals.
This is testing more Galois correspondence concepts. Here is an outline:
- Let E/F be a Galois extension with Galois group G.
- Suppose G is non-abelian.
- By the correspondence, if E/F were solvable by radicals, G would be solvable.
- But G is non-abelian, so it cannot be solvable.
- Therefore, E/F cannot be solvable by radicals.
This shows you can apply Galois correspondence to deduce field properties from group properties.
Tips for Preparing for Galois Theory Questions
Here are some suggestions for getting ready for technical Galois theory questions:
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Review core concepts like fields, groups, homomorphisms, normal subgroups, quotient groups, and basic Galois correspondence. Having this foundation locked down is essential.
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Work through examples of proving groups are or are not solvable, like showing S5 is not solvable. Get comfortable with these group theory arguments.
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Practice explaining concepts like the Fundamental Theorem out loud. You need to communicate complex ideas clearly and concisely.
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Anticipate definitional questions and learn terminology precisely. Interviewers may ask basic questions to test precise knowledge.
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Study proofs of key theorems like the Fundamental Theorem. Understand the logical flow and be able to explain the main steps.
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Brainstorm hypothetical questions and practice answering them under time pressure. Get used to thinking on your feet.
With diligent review and practice, you can master the Galois theory knowledge expected in interviews. The key is focusing on conceptual understanding and communication as much as derivations and proofs. Use these questions as an opportunity to demonstrate your passion for elegance and depth in mathematics!
Frequency of Entities:
field extensions: 5
splitting fields: 2
Galois groups: 5
solvability: 6
Fundamental Theorem of Galois Theory: 3
solvable groups: 4
soluble groups: 2
S5: 4
quintic: 2
radicals: 3
subfields: 2
subgroups: 3
Galois correspondence: 3
8 Answers 8 Sorted by:
I have now twice taught Galois theory to advanced high school students at PROMYS. This is a six week course, meeting four times a week, for students who already are comfortable with proofs and, in particular, have seen basic number theory. The second time, I taught the course as an IBL course, and you can read my worksheets here. Here is what I have done, and some thoughts about ways to do less.
To show that there isn’t a single quintic formula, I started both times with a weak version of the quintic being impossible to solve. This would make a very natural stopping point for a less ambitious course.
The general degree $n$ polynomial is written as $$x^n c_{n-1} x^{n-1} cdots c_1 x c_0 = (x-r_1) (x-r_2) cdots (x-r_n)$$ $$ Point out that $c_1$, . , $c_n$ are symmetric polynomials in the roots $r_1$, . , $r_n$. Show the formulas for quadratic, cubic, and (optionally) quartic equations and explain how they find the roots $r_i$ by expressing them in terms of the coefficients $c_i$ while staying inside the polynomials. One example is $r_1 = tfrac{-c_1 sqrt{c_1^2-4c_2}}{2}$. Since $c_1^2-4c_2 = (r_1-r_2)^2$, you can find the square root without leaving the world of polynomials. Teachers have taught them that square roots are always positive, so be ready for discussions about what you mean by “square root.” What you mean is any expression whose square is $c_1^2 – 4 c_2$. ).
Announce that your goal is to show, for $n geq 5$, that there is no expression for $r_1$, . , $r_n$ in terms of $c_1$, . use the operators $, $-$, $times$, $div$, and $sqrt[n]{ }$ on $c_n$ so that each $n$-th root stays inside the symmetric rational functions You should take the time to make sure that students understand the goal and know that it is not easy. This is a good way to get close to saying “there is no quintic formula.”
With this as the goal, define the symmetric group $S_n$ and note how it acts on formulas in $r_1$, …, $r_n$. Define the sign homomorphism $S_n to pm 1$ by the action of $S_n$ on $prod_{i
Now, define $F$ to be the set of all rational functions invariant for $A_n$. Clearly, $F$ is closed under $+$, $-$, $times$, $div$. Now the key Lemma: If $f in F$ is nonzero, and $f = g^n$ for some $g$ in $mathbb{C}(r_1, dots, r_n)$, then $sigma : tfrac{sigma(g)}{g}$ would be a group homomorphism $A_n to mathbb{C}^{ast}$. But we showed (for $n geq 5$) that there are no such homomorphisms! So $g$ must also be in $F$. Thus, our operations can never get us outside $F$, and in particular (for $n geq 5$) we cannot get to $r_1$, $r_2$, …, $r_n$. $square$
I really like this approach because it introduces so many key concepts — symmetries, characters, a set closed under the field operations, defining a subfield by its symmetries — without ever needing to define a field or a group as an abstract object. The first time, I did the above argument in a week of lectures and then went back to point out the key abstractions of “field”, “group” and “character” hiding in the proof. The second time, I did it in 3 weeks of IBL and introduced the group theory language explicitly as I went, but held back the definition of a field until we had completed the proof. Worksheet 9 is the climax.
As a side note, I never needed to prove the fundamental theorem of symmetric functions, though I assigned it as homework, and it is well-motivated by these results. The proof only needs the easy containment $mathbb{C}(c_1, ldots, c_n) subseteq mathbb{C}(r_1, ldots, r_n)^{S_n}$, not the equality.
Getting to abstract fields If you want to do anything harder than this, I think you need to define fields. Here, the fact that my students have already seen basic number theory is a huge advantage: They already know that, for $p$ a prime, every nonzero element of $mathbb{Z}/p mathbb{Z}$ is a unit, so they find it very quick to believe and prove the same thing about $k[x]/p(x) k[x]$ for $p(x)$ an irreducible polynomial.
There is a conceptual obstacle, though, which is convincing my students that they really can treat $mathbb{Q}[sqrt[3]{2}]$ as the same thing as $mathbb{Q}[x]/(x^3-2) mathbb{Q}[x]$. It takes them a long time to believe that, for example, knowing that $x^2+x+1$ is a unit in the ring $mathbb{Q}[x]/(x^3-2) mathbb{Q}[x]$ really proves that $tfrac{1}{sqrt[3]{2}^2+sqrt[3]{2}+1}$ is in $mathbb{Q}[sqrt[3]{2}]$. I think it is worth spending time to make them sit with this discomfort until they resolve it.
There is a reasonable half way goal to aim for here — that $sqrt[3]{2}$ cant be computed with $+$, $-$, $times$, $div$, $sqrt{ }$. That has the advantage of using field extensions, and the multiplicativity of degree, but not splitting fields or Galois groups. I talked about that more here.
To my surprise, in summer 2021, I never actually wound up needing to prove that the degree of a field extension was well defined! I talked about bases and spanning sets, and showed that $1$, $x$, …, $x^{deg(f)-1}$ was a basis for $k[x]/f(x) k[x]$, but I never proved that two bases of a field had the same cardinality or that degree was multiplicative! This was quite a relief; in summer 2018, I took a week away from the main material to do a crash course on linear algebra, and I lost a lot of momentum there.
If you are going for the full unsolvability of the quintic You need to introduce splitting fields and their automorphisms. At first, I found students swallow these with no hesitation, but they dont really realize what theyve said yes to. When you tell them that there is an automorphism of $mathbb{Q}[sqrt[3]{2}, omega]$ which maps $sqrt[3]{2}$ to $omega sqrt[3]{2}$, the students who dont just say yes to everything will start to feel very concerned. I think this is probably the point where it pays off to have really gotten them used to the notion that computations with polynomials really do prove things about concrete subfields of $mathbb{C}$.
The key lemma, from this perspective, is that if $f(x)$ is an irreducible polynomial in $F[x]$, and $K$ is a normal extension of $F$ in which $f$ has a root, then $f$ splits in $K$ and $text{Aut}(K/F)$ acts transitively on the roots of $f$ in $K$. From this, deduce:
Theorem Suppose that $f(x)$ is an irreducible polynomial of degree $n geq 5$ over $F_0$, let $K$ be a field in which $f$ splits and suppose that $text{Aut}(K/F_0)$ acts on the roots of $f$ by $S_n$. Then there is no tower of radical extensions $F_0 subset F_1 subset cdots subset F_r$ in which $f(x)$ has a root.
Its worth taking the time to convince students that this really is a rigorous and powerful formalization of “you cant solve the quintic with radicals”.
The key proof is to embed everything into a single splitting field $L$; let $G = text{Aut}(L/F_0)$. The details are on Worksheet 17, but the key point is that, on the one hand, we have a surjection $G to S_n$ but, on the other hand, we have a sequence of subgroups $G trianglerighteq G_0 trianglerighteq G_1 trianglerighteq cdots trianglerighteq G_r = { e }$ with each $G_{i+1}$ the kernel of a character $G_i to mathbb{C}^{ast}$, and this is impossible. (The warm-up version that I start the course with is easier because $G$ is $S_n$, so we are building the composition series in a concrete group we know, rather than a very abstract group where all we can say is that it surjects to $S_n$.)
There are a surprising number of things I didnt need to prove! I never introduced the notion of separability; the result is correct as stated for a normal inseparable extension. I also never proved the fundamental theorem of Galois theory! This proof starts with a chain of subfields and uses it to construct a chain of subgroups, but we never need the reverse construction! In particular, if there were two different subgroups that stabilized the same subfield, it would not effect the proof in any way.
I also never needed the abstract notion of a quotient group! I always had a concrete homomorphism $G to H$, and talked about its and kernel; I never needed to show that every normal subgroup was the kernel of a homomorphism. I put this on the homework, but it was never used in class.
Once you get here, a final issue is to construct an explicit example of a polynomial where the Galois group is $S_n$. The easy root is to take the polynomial $x^n + c_{n-1} x^{n-1} + cdots + c_1 x + c_0$ with coefficients in $mathbb{C}(c_1, ldots, c_n)$, since it is easy to see that $text{Aut}{big(} mathbb{C}(r_1, ldots, r_n)/mathbb{C}(c_1, ldots, c_n){big)}$ is $S_n$.
If you want to give an actual quintic with coefficients in $mathbb{Q}$ whose Galois group is $S_5$, there are various hacks to do this. What I did was to tell my students that, in 10 years, this would no longer be an issue: We will just write down a random polynomial $x^5 + c_4 x^4 + cdots + c_0$ with integer coefficients and roots $(r_1, ldots, r_5)$, compute the degree $120$ polynomial $g(x):= prod_{sigma in S_5} {big( }x-r_{sigma(1)} – 2 r_{sigma(2)} – cdots – 5 r_{sigma(5)} {big)}$, and check that the result is irreducible; this will prove that the polynomial has Galois group $S_5$. The need for a clever proof is simply because modern computers cant* compute and factor such large polynomials. I then did show them the clever proof that an irreducible quintic with two complex roots has Galois group $S_5$, which was the last result of the course.
* I might be wrong about this! I just attempted the case of a random integer monic quintic on my laptop. The first time, I didnt use enough working precision in computing the roots, and the degree $120$ polynomial didnt even even have real coefficients. But I went back and told Mathematica to use 100 digits for every floating point computation, and it got the polynomial in quite reasonable time, with every coefficient within $10^{-90}$ of an integer. This could make a genuine in class demo! Of course, I would need to know whether I am actually computing the degree $120$ polynomial correctly, but it seems unlikely that floating point errors would come out so near integers every time. Im not sure whether I can trust Mathematicas integer polynomial factorization for such large polynomials, but I know that Mathematicas routines for characteristic $p$ factorization are very good, and factoring my polynomial modulo the first $10$ primes finds a a degree $100$ factor modulo $3$; this is already enough to prove irreducibility. The future is now!
There is a nice book, specially written for high school students:
V. B. Alekseev, Abels theorem in problems and solutions. Based on the lectures of V. I. Arnold (to high school students), and also freely available online:
I recently came across a YouTube video by “not all wrong” called Galois-Free Guarantee! | The Insolubility of the Quintic that I think offers a lot of good ideas for making this material accessible to high school students. Despite the title, I disagree with the claim that its “Galois-Free”, and I dont think the material is handled with sufficient rigor, but I really like the numerical intuition it builds up to for how “permutation of roots” is a meaningful concept. I think you could take the same approach to establish why roots must be permutable, and turn this into a rigorous introduction to Galois theory.
“A Book of Abstract Algebra” by Charles Pinter gives a very accessible introduction to the topic, and it also covers Galois Theory. You can find it over here
Michio Kuga, Galoiss Dream was written with only a little bit of secondary school prerequisites.
I havent taught Galois theory to anyone, nor am I an expert in it. But as an interested bystander, I think its also useful to your young and enthusiastic cohort of students know that quintics as well as all polynomials can be solved by generalising the notion of a radical to an expression using elliptic modular forms and elliptic integrals in the case of a quintic, and in the general case, by an expression involving Siegel modular forms and hyper-elliptic integrals. A formula for the roots was given by Thomae in 1870. Uemara gave a simpler formula in terms of theta constants in 1984.
I appreciate that this is far removed from Galois theory but it is not far removed from solving the quintic. It might be worth discussing this when giving the historical context of Galois theory. And in the same discussion it might be worth outlining that there is a differential Galois theory that studies the Galois groups of differential equations and which gives criteria when a differential equation can be solved analytically in terms of elementary functions.
Rather than the insolubility of the quintic, I would focus on the ways in which group theory can help you solve equations.
- Indicate how solutions of linear equations carry a group action of a vector group (Edit: since the solutions form a translate of a vector subspace of the domain).
- Indicate how solutions of a quadratic equation carry a group action.
- Show how you can derive the construction of the pentagon using group theory.
- Indicate how this can be generalised to the construction of the 17-gon (via a succession of quadratic equations).
- Perhaps indicate how, if you could construct the root of a cubic (using a straightedge and a compass), you can construct a heptagon.
- Indicate the solution of the cubic and quartic equations using group theory.
- Note that these solutions use solvability of the group. Hence, indicate how solvability of the group is a key idea.
Please take a look at Math Girls 5: Galois Theory, which I translated and was published just last month. If you arent familiar with the series, these books, originally published in Japan, are novels in that they present fictional characters (high school students who explore various topics in math for fun) and have a thin veneer of plot, but most of the content is mathematics.
MG5 isnt a comprehensive presentation of Galois Theory, but rather focuses on Galois original paper regarding solutions of equations. The bulk of that comes in the final chapter, with preceding chapters introducing permutations, symmetric polynomials, and other requisite information. Theres a lengthy fan review of the book on Amazon.com giving a much more detailed description of the content, if youre interested. You can also use Amazons “Look Inside” feature to browse the table of contents.
Because these books are written with the aim of being accessible to motivated high school students, perhaps they will give you some ideas regarding how to present this challenging content?
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Galois Theory Explained Simply
FAQ
What is Galois theory useful for?
What is the essence of Galois theory?
What is an example of Galois theory?
What did Galois prove?
How do I learn Galois theory?
There is more preliminary work than you might guess. You could take an entire abstract algebra course, and when you were done, you would be ready to begin Galois theory. You need some group theory. An explanation why the group A5 A 5 is a “simple group.” And an introduction to fields, and you are ready to start to tackle Galois theory.
Does Galois theory have a formal background?
No formal background in Galois Theory is assumed. While a complete proof of the Fundamental Theorem of Galois Theory is given here, we do not discuss further results such as Galois’ theorem on solvability of equations by radicals. An annotated list of references for Galois Theory appears at the end of Section 5.
What is the goal of Galois theory?
The goal of the book is described in the original preface. In a few words it can be sketched as follows: Galois theory is presented in the most elementary way, following the historical evolution. The main focus is always the classical application to algebraic equations and their solutions by radicals.
Where can I find a list of references for Galois theory?
An annotated list of references for Galois Theory appears at the end of Section 5. By way of references for the last section, viz., Norms and Traces, we recommend Van der Waerden’s “Algebra” (F. Ungar Pub. Co., 1949) and Zariski–Samuel’s “Commutative Algebra, Vol. 1” (Springer-Verlag, 1975).
