Some Facts About The Ring of Algebraic Integers
Point of post: In this post we’ll discuss some very basic facts concerning the ring of algebraic integers that shall become useful in other posts.
We’ve previously discussed the set of all algebraic numbers at least to the extent to show that the set of all of them is countable. In this post we shall restrict our attention to the algebraic integers, which are basically the result of considering algebraic numbers which are roots of monic polynomials. Our goal in this post is to give some alternate characterization of the algebraic integers in terms of integral matrices, show they are a subring of , and show that an eigenvalue of a matrix with algebraic integer entries must be itself an algebraic integer.
Definitions and Basics
Let denote, as usual, the polynomial ring over . We call a real number an algebraic integer if there exists some monic such that . We denote the set of all algebraic integers by .
Next, as usual, let denote the set of all matrices with integer entries. Define then to be equal to . We call an element of an integral matrix.
Our first theorem is an interesting one. Indeed, for a matrix let denote it’s spectrum then:
Proof: We merely note that if for some then is monic (the characteristic polynomial of ) and . Thus, .
Conversely, let , then by definition there exists some monic such that . Let denote the companion matrix of . Then, by common knowledge we have that and so in particular so that . Since the conclusion follows.
With this it’s not difficult to prove that is a subring of . Indeed
Theorem: is a subring of .
Proof: Let . Then, by our previous theorem there exists where are of size respectively such that and . But, this then implies (by basic matrix analysis) that , , and where denotes the Kronecker product. The conclusion then follows from the previous theorem.
Our final goal in this very short introduction to the ring of algebraic integers is to show that eigenvalues of a matrix whose entires are algebraic integers is itself an algebraic integer. But first, a lemma
Lemma: Let . Then, there exists some , some , and such that for every .
Proof: Since there exists matrices and for which . Let then where in general (and the case for more than two elements in the tensor product is defined inductively). Then, if where here denotes the usual Kronecker product then we note that where we’ve used the common fact that with these definitions
The conclusion follows.
We now prove the pivotal part of this post. Namely that if is a matrix with entries in then . More formally:
Theorem: Let for any . Then, .
Proof: Let where by assumption for every . Now, by the previous lemma we may find some , a , and matrices such that . Let be the matrix with a in the entry and elsewhere. Then, let denote the matrix
Then, for every let be the associated eigenvector. Then,
and since we have that . Since was arbitrary the conclusion follows.
One last thing to notice is that . More precisely:
Theorem: Let be rational algebraic integer. Then, is an integer.
Proof: Suppose that and let is the polynomial such that . Then, a quick check shows that this implies but since and we may conclude that .
1.Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.