## 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:
• Generalities about arithmetic of finite normal extensions of number fields and function fields
• More detailed study of the Galois groups of extensions of the p-adic field Qp and other ``local fields''
• The Kronecker-Weber theorem: every abelian extension of Q is contained in a cyclotomic extension.
• In particular, every quadratic extension of Q is contained in a cyclotomic extension. This can be used to prove Quadratic Reciprocity and some generalizations.
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

• Galois theory of Riemann surfaces: covering maps as field extensions, Galois groups and fundamental groups of punctured Riemann surfaces. Generalizes ideas introduced in PS2, problem 5; PS4, problem 7; and the example of isogenies between elliptic curves over C.
• Galois theory and algebraic geometry: Some of the same ideas in the algebraic setting, including varieties of degree >1. Again we saw some of these ideas in the setting of elliptic curves and of PS2, problem 5; also, the Galois group of the ``general polynomial'' can be interpreted geometrically in terms of the quotient of n-space with coordinates x1,...,xn by Sn.

### 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

• Elimination theory: resultants, etc. How to actually compute things like the minimal polynomial of x+y where f(x)=g(y)=0, or where x,y are distinct roots of f(x)?
• Resolvents and the determination of Galois groups. How to determine the Galois group of a given polynomial and exhibit the subfields promised by the Galois correspondence?
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.