Number Theory or Complex Variables?

[acme37]

Hey all, great site and I look forward to contributing more. For now, a question...

For next semester I need to choose between Number Theory or Complex Variables. I am under the impression that complex variables will be the more useful class for my physics education, however I had some concerns about the level of difficulty, already being registered for 15 hours. Further, number theory is now listed as a "recommended" prerequisite for Abstract algebra, which I plan on taking next fall. But, I'm not sure exactly how what I would learn in the Number Theory course would apply to what I will be studying in physics. Are some of the number theory topics things I can pick up on my own? Are complex variables tough? What is the more useful class? Here are the course descriptions:

221. Theory of Numbers. The Euclidean algorithm, Euler’s phi function, simple continued fractions, congruences, Fermat’s theorem, Wilson’s theorem, and elementary Diophantine equations.

261. Complex Variables. Study of complex numbers, analytic and elementary functions, transformations of regions, properties of power series, including Taylor’s and Laurent’s. The calculus of residues with applications, conformal mapping with emphasis upon boundary value applications

 

[tmc]

Personnally, I would go with complex variables 1&2 instead of number theory and abstract algebra.

 

[jcsd]

In terms of usefulness for physics complex variables wins hands down, it contains stuff that is prequiste in order to study quantum mechanics. On the other hand many people would spend a long time scratching their head before they could think of one application of number theory in physics.

Also I wouldn't be worried at all about not taking number theory before taking abstarct algebra.

 

[mathwonk]

most important resilts in number theory use compelx analysis or abstract algebra in their proofs, e.g. Wiles' proof of Fermat's last theorem, Weil's proof of the Artin-Riemann hypothesis for curves over finite fields.

Although "elementary" proofs of the latter e.g. have been produced by Stepanov and others, recent treatements of them require hundreds of pages as opposed to the 1 or 2 pages needed for proofs using abstract algebraic geometry of surfaces.

 

[mathwonk]

let me explain slightly the last comment. the riemann hypothesis for varieties over finite fields is an estimate for the number of points that have coordinates in a given finite base field containing the coefficients.The "frobenius" mapping, i.e. merely raising a number to the qth power, is the identity on precisely the elements of the finite field with q elements. hence counting the number of points with coordinates in that field is the same as counting the number of "fixed points" of the frpbenius map.

according to the method of lefschtetz in topology, the unmber of fixed points of a map is best counted by counting the number of intersections of its graph with the diagonal, i.e. the graph of the identity map. Thus to prove the riemann hypotheses for a curve C, means to estimate the intersection number of the graph of the frpobenius map with the diagonal on the surface CxC.

This is achieved by the Castelnuovo inequality, provable by the easy "Riemann Roch" theorem for surfaces, the last result in the book of kempf, algebraic varieties. (The riemann hypothesis is achieved in the previous section, in three pages, even without RRT.)

 

[acme37]

Thanks for the advice. I've decided to take Complex Variables. I bought two dover books on Number Theory which I am working through this Thanksgiving Break

Cheers

 

[Kalimaa23]

Good choice. I've never taken a number theory course, although my abstract algebra course did indeed use some of the topics you mentioned. Don't worry, you'll be fine. The complex analysis course however is something you will dearly need, wathever you do in physics.