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Given that a math major won't just be getting more practice at the sorts of math encountered in high school, what is it about?  Here's a list of some of the major fields that are the foundation of a mathematics education.   

Given that a math major won't just be getting more practice at the sorts of math encountered in high school, what is it about?  Here's a list of some of the major fields that are the foundation of a mathematics education.   

<i>For each field, I hope give a very general description and to describe a few problems/theorems that are nice results and capture some of the essence of a subject.  I'll aim for a few paragraphs about each one.  Then I will give a problem or two that is a much more open-ended question that has motivated the development of large amounts of theory.  For now I have no written about any of these -- please leave feedback and suggestions on the talk page and I will try to arrive at the best set of motivating problems!</i>

<i>For each field, I plan to give a very general description and to describe a few problems/theorems that are nice results and capture some of the essence of a subject.  I'll aim for a few paragraphs about each one.  Then I will give a problem or two that is a much more open-ended question that has motivated the development of large amounts of theory.  For now I have no written about any of these -- please leave feedback and suggestions on the talk page and I will try to arrive at the best set of motivating problems!</i>

=== Algebra ===

=== Algebra ===

# The ancient Greeks were fascinated by geometric constructions with straightedge and compass.  They could bisect angles and take square roots, but they couldn't trisect angles or take cube roots.  Why not?

# The ancient Greeks were fascinated by geometric constructions with straightedge and compass.  They could bisect angles and take square roots, but they couldn't trisect angles or take cube roots.  Why not?

A motivating question: failure of unique factorization in certain number fields (with explicit examples)

The development of the modern field of algebra has been deeply influenced by research in number theory.  An example of a problem from number theory that has provided far-reaching motivation for work in pure algebra has been the question of unique factorization.

 

It's a well-known fact, the [[fundamental theorem of arithmetic]], that every integer can be factored in a unique way as the product of prime numbers.  It's natural to wonder whether this unique factorization holds in other simple rings, for example, the set of complex numbers whose real and complex parts are both integers (for example, <math>3+7i</math>).  These are the so-called Gaussian integers, and there is a good notion of "prime", and that every Gaussian integer can be uniquely factored into primes!  For example, 7+0i is a Gaussian prime, while 5+0i is not: <math>5+0i</math> factors as <math>(2+i)(2-i)<math>, and both <math>2-i</math> and <math>2+i</math> are prime.  In general, it turns out that

a Gaussian integer <math>a+bi</math> is prime if either one of <math>a</math> or <math>b</math> is zero, and the other is a prime congruent to three mod 4, or if <math>a</math> and <math>b</math> are nonzero, and <math>a^2+b^2</math> is a prime (in the usual sense).

 

So unique factorization works well for the Gaussian integers, but what about other settings?  An example similar to the Gaussian integers is the set of real numbers of the form <math>a+b\sqrt{7}</math>: note that <math>(a+b\sqrt{7})(c+d\sqrt{7}) = (ac+7bd)+(ad+bc)\sqrt{7}</math>, so the numbers of this form are closed under multiplication.  Here too, and in many other cases, there is a good notion of unique factorization.  It was formerly assumed that <math>\mathbb Z[\sqrt{d}]</math> would have unique factorization, but this turns out not to be the case.  Consider the ring of complex numbers of the form <math>a+b\sqrt{-5}</math>.  We can write <math>6 = 2 \cdot 3 = (1-\sqrt{-5})(1+\sqrt{5})</math>.  A bit more work shows that both of these are prime factorizations, and they're not equivalent!  This discovery necessitated the careful investigation of unique factorization, a purely algebraic notion that had not been seriously considered.  Ernst Kummer proposed a proof of [[Fermat's Last Theorem]] which failed only because it assumed some properties of unique factorization, but further consideration of this problem led to a revolution in algebra.

=== Linear Algebra ===

=== Linear Algebra ===