Further directions (and potential paper topics) in Galois theory

NB: Some of these topics are interrelated (even among different sublists); some are interactions of Galois theory with other branches of mathematics and require tools outside Math 250a and its prerequisites.

Galois theory and algebraic number theory

Galois theory is an important tool for studying the arithmetic of ``number fields'' (finite extensions of Q) and ``function fields'' (finite extensions of Fq(t)). In particular: One source is Number Fields by Daniel Marcus (New York: Springer, 1977), QA247.M346 in Cabot. As you might guess from the title, function fields aren't the main focus, but most of the results carry over with little change.

Galois theory and geometry

More about p-polynomials: Dickson invariants, ``linearized algebra'', etc.

Just as the general polynomial has Galois group Sn, the general p-polynomial has Galois group GLn(Fp). The elementary symmetric functions correspond to ``Dickson invariants'' of the action of GLn(Fp) on polynomials in n variables over Fp. Other aspects of the theory of F[X] require more interesting modifications in the setting of p-polynomials, since polynomial multiplication is replaced by composition of p-polynomials, which is not commutative!

Differential Galois theory

Analogous to the algebraic theory of polynomial equations Xn+a1Xn-1+...+an-1X+an=0 over a field is an algebraic theory of linear differential equations y(n)+a1y(n-1)+...+an-1y'+any=0 over a differential field. In this context we again have field extensions, normal closures, and even differential Galois groups, a differential Galois correspondence, and solvability criterion. The roles played in classical Galois theory by [E:F] and the finite subgroup Gal(K/F) of Sn are assumed by the trascendence degree and (usually) a Lie subgroup of GLn -- indeed, differential Galois theory was Lie's original motivation. For instance, Bessel functions (except those of half-integral order) cannot be expressed in terms of elementary functions and their integrals because the Bessel differential equation has differential Galois group containing SL2(R) which is not solvable! Naturally, I cannot assume extensive background in Lie theory, because that will be a major topic of Math 250b; but you should at least be comfortable thinking about groups like GLn if you want to take this on.
Notes on differential algebra and differential Galois theory, in PS and PDF

Computational issues

See Henri Cohen's A Course in Computational Algebraic Number Theory (Berlin: Springer, 1993 = Graduate Texts in Mathematics #138), QA247.C55 in Cabot.

Approaches to the inverse Galois problem

Constructions and results concerning polynomials with prescribed Galois groups. As noted on the Math 250 homepage, a good starting text here is J.-P. Serre's Topics in Galois Theory (Boston: Jones & Bartlett, 1992), QA214.S47 in Cabot.