A number is an "algebraic integer" if it is the root to a monic polynomial with integer coefficients. Artin says (Algebra, p. 411):
The concept of algebraic integer was one of the most important discoveries of number theory. It is not easy to explain quickly why it is the right definition to use, but roughly speaking, we can think of the leading coefficient of the primitive irreducible polynomials $f(x)$ as a "denominator." If $\alpha$ is the root of an integer polynomial $f(x)=dx^n+a_{n-1}x^{n-1}…$ then $d\alpha$ is an algebraic integer, because it is a root of the monic integer polynomial $x^n + a_{n-1}x^{n-1} + … + d^{n-1}a_0$.
Thus we can "clear the denominator" in any algebraic number by multiplying it with a suitable integer to get an algebraic integer.
When I first learned of algebraic integers, I looked online and saw some hints that maybe they were used to prove the Abel-Ruffini theorem. So I put off questioning their usage for a while; I now think I understand one proof of this theorem (the one at the end of Artin's Algebra) and it has nothing to do with algebraic integers (that I can tell).
So basically: why is it important if a number is an algebraic integer? I think I understand what he's saying about the relationship between roots of integer polynomials and algebraic integers, but I fail to see why this is "one of the most important discoveries of number theory."
Best Answer
Suppose that we desire to consider as "integers" some subring $\:\mathbb I\:$ of the field of all algebraic numbers. To be a purely algebraic notion, it cannot distinguish between conjugate roots, so if $\rm\:\alpha,\alpha'$ are roots of the same polynomial irreducible over $\rm\:\mathbb Q\:,\:$ then $\rm\:\alpha\in\mathbb I\iff \alpha'\in\mathbb I\:.\:$ Also we desire $\rm\:\mathbb I\cap \mathbb Q = \mathbb Z\ $ so that our notion of algebraic integer is a faithful extension of the notion of a rational integer. Now suppose that $\rm\:f(x)\:$ is the monic minimal polynomial over $\rm\:\mathbb Q\:$ of an algebraic "integer" $\rm\:\alpha\in \mathbb I\:.\:$ Then $\rm\:f(x) = (x-\alpha)\:(x-\alpha')\:(x-\alpha'')\:\cdots\:$ has coefficients in $\rm\:\mathbb I\cap \mathbb Q = \mathbb Z\:.\:$ Therefore the monic minimal polynomial of elements $\in\mathbb I\:$ must have coefficients $\in\mathbb Z\:.\:$ Conversely, one easily shows that the set of all such algebraic numbers contains $1$ and is closed under both difference and multiplication, so it forms a ring. Moreover, as Artin's quote shows, the quotient field of $\rm\:\mathbb I\:$ is the field of all algebraic numbers. Hence a few natural hypotheses on the notion of an algebraic integer imply the standard criterion in terms of minimal polynomials.
Because this notion of integer faithfully extends the notion of rational integers, we can employ algebraic integers to deduce results about rational integers. This often results in great simplifications because many diophantine equations become simpler - being "linearized" - when one factors them in algebraic extension fields. For example, see proofs about Pythagorean triples using Gaussian integers, or classical proofs of FLT for small exponents employing algebraic integers.