Galois Theory Primer
August, 2000

This page is not an official publication of 
the University of North Texas.

CLASS MEETS:   August 1 - August 17,  TWR 12:30 - 1:30, GAB 318

TEXT:   Abstract Algebra (6th edition), John B. Fraleigh

OFFICE HOURS:   Any time you can find me.  Feel free to make an appointment.

This August I will give an informal seminar on Galois theory.  The goal will be to prepare for the algebra qualifying exam, but the lectures will be open to everyone, and no registration is necessary.  We will meet somewhere in the GAB, two or three times a week, at a time to be determined.  We will probably begin on Monday, July 31, and go on for three weeks.  If you are interested, please let me know at what times and days of the week would be best for you.  I will post information here, but I will be out of town until just before the course begins, so email will be the only way to reach me.  Once I am back, feel free to stop by my office to ask questions any time.  You can make an appointment in class, by email, or at x-3326.

I intend to use Abstract Algebra, by J. Fraleigh.  We will begin with a brief review of Sections 8.1 (extension fields) and  8.2 (vector spaces).  Then we will go over 8.3 (algebraic extension fields), after which we will jump to Chapter 9.  We will cover as much as possible of  9.1 (field automorphisms),  9.2 (extensions of field isomorphisms),  9.3 (splitting fields),  9.4 (separable extensions),   9.6 (Galois theory),  9.7 (examples), and  9.8 (cyclotomic extensions).  If time permits, we will also cover  8.4 (straight-edge and compass constructions),  8.5 (finite fields), and  9.9 (quintic equations).  Below are some Galois theory problems, in TeX (until someone does me the favor of teaching me how to put math on the web!), which we will try to solve as we go.  All but the last are from old UNT qualifying exams.

1.  Find the Galois group of $\Bbb Q(\sqrt2,\sqrt3,\sqrt5)$ over $\Bbb Q$, and state the fundamental theorem of Galois theory.

2.  Let $F\subset E$ be fields, and let $\alpha\in E$ be algebraic over $F$ of odd degree.  Prove that $\alpha^2$ is also algebraic over $F$, and $F(\alpha^2) = F(\alpha)$.

3.  Prove that any polynomial over any field has a splitting field.

4.  Compute the Galois group of $x^3-2$ over $\Bbb Q$.

5.  Prove that $\Bbb Q(i+2^{1/4})$ is the splitting field of $x^4-2$ over $\Bbb Q$.

6.  Find the Galois group of $g(x)= x^3+2x^2+4x-2$ over $\Bbb Q$.

7.  Find the splitting field of $f(x) =x^5-2x^4-3x^3-x^2+2x+3$ over both $\Bbb Q$ and $\Bbb Z_3$.

8.  Find polynomials in $\Bbb Q[x]$ whose Galois groups over $\Bbb Q$ are $C_2$, $C_3$, and $S_3$, respectively.

9.  Prove that $\pi/3$ is not trisectable with a straight-edge and compass.

10.  Suppose that $f\in\Bbb Q[x]$ is \irr, of prime degree~$p$, and has only~2 real roots in $\Bbb C$.  Prove that its Galois group
is $S_p$.

11.  Let $K$ be a field of characteristic~0, and let $L$ be a Galois extension of $K$ whose Galois group is $C_p$, for some prime~$p$.  Prove that $L =K(\alpha)$ for some $\alpha \in L$ such that $\alpha^p \in K$ iff $K$ contains a primitive $p^{th}$-root of unity.

12.  Let $F$ be the finite field of order $p^{12}$ for some prime~$p$.  Prove that $\sigma(x)=x^p$ is a field automorphism of $F$.  Prove that the Galois group of $F$ over $\Bbb Z_p$ is cyclic, and generated by $\sigma$.  Use the Galois correspondence to describe all subfields of $F$.

13.  Find the Galois group of $f(x)$ over the finite field $\Bbb F_q$, given that $f$ is the product of three irreducible polynomials of degrees~6, 10, and~15.