Algebraic number theory - Wikipedia
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers , finite fields , and function fields .
Algebraic numbers In this chapter we introduce the basic objects of the course. 1.1. Algebraic numbers. Minimal polynomials. Definition 1.1.1. A complex number is algebraic if it is the solution to some polynomial equation with coe cients in Q. The set of all algebraic numbers is denoted by Q. Examples. Every rational is algebraic, as are i, p
Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and units in the ring of integers, the extent to which unique factorization holds, and so on.
Number Theory is one of the oldest and deepest Mathematical disciplines. In the broadest possible sense Number Theory is the study of the arithmetic properties of Z, the integers.
A Textbook of Algebraic Number Theory | SpringerLink
2022年4月26日 · Serves as a comprehensive textbook of algebraic number theory; Discusses proofs of almost all significant basic theorems of algebraic number theory; Presents numerous solved examples, exercises and problems in every chapter
Algebraic Number Theory | Brilliant Math & Science Wiki
Algebraic number theory is the study of roots of polynomials with rational or integral coefficients. These numbers lie in algebraic structures with many similar properties to those of the integers.
2022年5月21日 · An algebraic integer is a root of a monic polynomial in Z[x]. An algebraic number is the root of any non-zero polynomial in Z[x]. A number field is a finite extension of Q. If K, L are fields and K ⊆ L, we say that L is an extension field of K and K is a subfield of L. We write [L : K] = dimK L, the dimension of L over K. Example.
The goals of the subject include: (i) to use algebraic concepts to deduce information about integers and other rational numbers and (ii) to investigate generaliza- tions of the integers and rational numbers and develop theorems of a more general nature.
The lecture builds on the lecture Commutative Algebra from the previous term, the lecture notes of which are available on http://maths.pratum.net. The lecture provides an introduction to the most basic classical topics of (global) algebraic number
Algebraic number theory F.Beukers February 2011 1 Algebraic Number Theory, a crash course 1.1 Number elds Let K be a field which contains Q. Then K is a Q-vector space. We call K a number eld if dimQ(K) < ∞. The number dimQ(K) is called the degree of the number field. Notation: [K: Q].