The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
Definition
Prerequisites
The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted Q, together with its usual operations of addition etc.
Another notion needed to define algebraic number fields is vector spaces. To the extent needed here, vector spaces can be thought of as consisting of sequences (or tuples)
whose entries are elements of a fixed field, such as the field Q. Any two such sequences can be added by adding the entries one per one. Furthermore, any sequence can be multiplied by a single element c of the fixed field. These two operations known as vector addition and scalar multiplication satisfy a number of properties that serve to define vector spaces abstractly. Vector spaces are allowed to be "infinite-dimensional", that is to say that the sequences constituting the vector spaces are of infinite length. If, however, the vector space consists of finite sequences
The rational numbers, expressed as the quotient of two integersa and b, b not equal to zero, satisfy the above definition because x = a/b is the root of bx − a.
The quadratic surds (irrational roots of a quadratic polynomial ax2 + bx + c with integer coefficients a, b, and c) are algebraic numbers. If the quadratic polynomial is monic (a = 1) then the roots are quadratic integers.
The constructible numbers are those numbers that can be constructed from a given unit length using straightedge and compass. These include all quadratic surds, all rational numbers, and all numbers that can be formed from these using the basic arithmetic operations and the extraction of square roots. (Note that by designating cardinal directions for 1, −1, i, and −i, complex numbers such as 3 + √2i are considered constructible.)
... mathematics (including number theory, abstract algebra and others), and applications to fields outside of mathematics (including computer science, optimization, operations research and engineering).
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Kent County Daily Times
17 Oct 2024
Kent County Daily Times
17 Oct 2024