What branch of Mathematics does the study of Algebraic/Transcendental Numbers lie in?

Question

I've always been fascinated by polynomials, ever since first learning them in high school. I absolutely adore the notion of 'playing around with the coefficients' and watching what happens to the location of the roots, or how we can construct bounds on the roots using just the coefficients.

For example, the Eneström–Kakeya Theorem is particularly pretty to me.

I will be absolutely delighted the day that somebody proves that $e+\pi$ is irrational (which it 'probably' is).

I was wondering, does the close study of polynomials, algebraic/transcendental numbers fall under any specific branch of Mathematics?

basically anything that deals with numbers is studied in number theory — alkabary, May 27 '15 at 05:43
You may wish to check the wikipage on transcendence theory, which begins: "Transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways." — Benjamin Dickman, May 27 '15 at 06:30

score 9 · Accepted Answer · 2015-05-27T05:32:09.563

What you're describing is number theory, which takes from all kinds of other areas such as abstract algebra, analysis, computer science, and tons of other areas.

Edit, for examples of how each area is important.

Abstract Algebra

The set of algebraic numbers forms a field, which means they can be added and multiplied and still stay algebraic.

You can generalize the idea of an "algebraic number" to an abstract algebraic "element" (http://en.wikipedia.org/wiki/Algebraic_element).

Real Analysis

The "Lebesgue Measure" or size of the algebraic numbers is $0$, which basically means that almost all real numbers are NOT algebraic.

Computer Science

Algebraic numbers are computable, which means that there exists an algorithm to actually well, compute them. Algorithms are a part of computer science.

Number theory deals with ALL these areas, and deals with much more than just algebraic or transcendental numbers.

Cheers for that. So within Number Theory, is there a 'sub-branch' that deals specifically with transcendentals etc? For example, within Abstract Algebra, you've got Hecke Algebras, Lie Algebras etc. — Trogdor, May 27 '15 at 05:59
@Trogdor, there is indeed a subject within number theory called "transcendental number theory." It is concerned with transcendence of special values of functions like exponential, elliptic, or modular functions. — KCd, May 27 '15 at 06:18

What branch of Mathematics does the study of Algebraic/Transcendental Numbers lie in?

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