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Integral Extensions 2.1 Integral Elements 2.1.1 Definitions and Comments 2.1.2 Lemma
| Content Provider | Semantic Scholar |
|---|---|
| Abstract | Let R be a subring of the ring S, and let α ∈ S. We say that α is integral over R if α is a root of a monic polynomial with coefficients in R. If R is a field and S an extension field of R, then α is integral over R iff α is algebraic over R, so we are generalizing a familiar notion. If α is a complex number that is integral over Z, then α is said to be an algebraic integer For example, if d is any integer, then √ d is an algebraic integer, because it is a root of x 2 − d. Notice that 2/3 is a root of the polynomial f (x) = 3x − 2, but f is not monic, so we cannot conclude that 2/3 is an algebraic integer. In a first course in algebraic number theory, one proves that a rational number that is an algebraic integer must belong to Z, so 2/3 is not an algebraic integer. There are several conditions equivalent to integrality of α over R, and a key step is the following result, sometimes called the determinant trick. Let R, S and α be as above, and recall that a module is faithful if its annihilator is 0. Let M be a finitely generated R-module that is faithful as an R[α]-module. Let I be an ideal of R such that αM ⊆ IM. Then α is a root of a monic polynomial with coefficients in I. n j=1 (δ ij α − c ij)x j = 0, 1 ≤ i ≤ n. In matrix form, we have Ax = 0, where A is a matrix with entries α − c ii on the main diagonal, and −c ij elsewhere. Multiplying on the left by the adjoint matrix, we get ∆x i = 0 for all i, where ∆ is the determinant of A. But then ∆ annihilates all of M , so ∆ = 0. Expanding the determinant yields the desired monic polynomial. ♣ |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.math.uiuc.edu/~r-ash/ComAlg/ComAlg2.pdf |
| Alternate Webpage(s) | https://faculty.math.illinois.edu/~r-ash/ComAlg/ComAlg2.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
