By Tom Weston
Read or Download Algebraic Number Theory [Lecture notes] PDF
Similar number theory books
This is often the vintage introductory graduate textual content. center of the ebook is degree conception and Lebesque integration.
A paean to 20th century research, this contemporary textual content has a number of vital issues and key gains which set it except others at the topic. an enormous thread all through is the unifying impression of the idea that of absolute continuity on differentiation and integration. This ends up in basic effects reminiscent of the Dieudonné–Grothendieck theorem and different complicated advancements facing vulnerable convergence of measures.
This ebook has grown out of a process lectures i've got given on the Eidgenossische Technische Hochschule, Zurich. Notes of these lectures, ready for the main half by way of assistants, have seemed in German. This e-book follows a similar common plan as these notes, notwithstanding widespread, and in textual content (for example, Chapters III, V, VIII), and in awareness to aspect, it is extremely diversified.
- Class Field Theory
- Elementary Number Theory in Nine Chapters, Second Edition
- Introduction to Numerical Analysis english
- Problem-Solving and Selected Topics in Number Theory: In the Spirit of the Mathematical Olympiads
Extra info for Algebraic Number Theory [Lecture notes]
The correct condition turns out to be the following. 1. Let R be an integral domain contained in some field K. An element α ∈ K is said to be integral over R if it satisfies some monic polynomial in R[x]. R is said to be integrally closed in K if every element in K which is integral over R actually lies in R. Note that the definition says nothing about monic polynomials in R[x] actually having roots in K; it says only that if they do have roots in K, then these roots lie in R. Note also that there is nothing about the minimal polynomial of α in the definition; any monic polynomial at all will do, irreducible or not.
There are three possibilities for the factorization of f¯(x) in Fp [x]. First of all, f¯(x) could be irreducible. Second, f¯(x) could factor as a product of distinct, monic linear (and therefore irreducible) polynomials. Third, f¯(x) could factor as the square of a single monic linear polynomial. We will consider all three cases separately. Suppose first that f¯(x) is irreducible in Fp [x]. Then Fp [x]/(f¯(x)) is a field, so OK /pOK is as well. pOK is therefore a prime ideal of OK , by the definition of prime ideal, so it does not factor any further.
The problem with R is that it is missing certain elements; we will see the full solution in the next section. 2. 1. Integrally closed rings. The key to our search for the right special subring of a number field K is the “good factorization theory” condition. As we have seen, it is unreasonable to expect unique factorization, although there is still some hope that we may be able to get unique factorization of ideals. What we need, then, is some condition which √ is weaker than UFD but still strong enough to eliminate the problem case of Z[ 3].