I'm currently an undergraduate student in mathematics. I am currently taking Algebra. The course is interesting, but I have grown very curious about the usefulness of algebra.
I am NOT asking about "applications of algebra to real-life". I am asking about how algebra can be used to solve math problems. Unfortunately, Googling "applications of algebra" is not all that helpful.
Right now I can only recall seeing two instances of "useful" applications of algebra — a proof of Fermat's Little Theorem, and determining whether a polynomial is solvable in radicals by looking at its Galois group.
What interests me about both problems is that they are of interest to someone who has not necessarily encountered abstract algebra yet (e.g. what is the remainder when you divide k^p by p? can you write explicitly the roots of some polynomial using only the integers and the specified functions?).
At least from the way my course is currently progressing, it feels as though such applications are few and far in between. We are currently making observations about permutations (e.g. if p and q are permutations, then pq and qp have "similar forms"), which is interesting, but I fail to see how algebra has helped make any interesting deduction — all interesting results so far about permutations (e.g. the one mentioned above) were all done without any algebraic result.
Only when we ask a question using algebraic terminology was algebra required (e.g. show An is a normal subgroup of Sn). If algebra were only used to answer questions about algebra, there would be no real need to study algebra, right?
What are some other "elementary" applications of algebra? What are some other interesting results I would be able to understand after an introductory course?
I have a suspicion that finding answers to these questions would better my understanding of algebra, but I have had difficulty in finding many good answers.
Best Answer
One of the most important results you learn in a first course on abstract algebra is Burnside's lemma, which has many applications in combinatorics and number theory. Some time ago I wrote a series of blog posts leading up to a powerful corollary of Burnside's lemma called the Polya enumeration theorem (including several applications, so you should look in those posts for them), which can be used to count many things; it was originally used to count chemical compounds.
The Polya enumeration theorem in turn can be used to prove a powerful result in combinatorics called the exponential formula, which gives you an enormous amount of information about permutations. For example, using the exponential formula you can prove results like
relatively easily.