Algebraic number field in the context of "Field extension"

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

The best known fields are the field of rational numbers, the field of real numbers, and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements.

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. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

In mathematics, the ring of integers of an algebraic number field ${\displaystyle K}$ is the ring of all algebraic integers contained in ${\displaystyle K}$. An algebraic integer is a root of a monic polynomial with integer coefficients: ${\displaystyle x^{n}+c_{n-1}x^{n-1}+\cdots +c_{0}}$. This ring is often denoted by ${\displaystyle O_{K}}$ or ${\displaystyle {\mathcal {O}}_{K}}$. Since any integer belongs to ${\displaystyle K}$ and is an integral element of ${\displaystyle K}$, the ring ${\displaystyle \mathbb {Z} }$ is always a subring of ${\displaystyle O_{K}}$.

The ring of integers ${\displaystyle \mathbb {Z} }$ is the simplest possible ring of integers. Namely, ${\displaystyle \mathbb {Z} =O_{\mathbb {Q} }}$ where ${\displaystyle \mathbb {Q} }$ is the field of rational numbers. And indeed, in algebraic number theory the elements of ${\displaystyle \mathbb {Z} }$ are often called the "rational integers" because of this.

In mathematics, certain subsets of some fields are called orders. The set of integers is an order in the rational numbers (the only one). In an algebraic number field ⁠${\displaystyle K}$⁠, an order is a ring of algebraic integers whose field of fractions is ⁠${\displaystyle K}$⁠, and the maximal order, often denoted ⁠${\displaystyle {\mathcal {O}}_{K}}$⁠, is the ring of all algebraic integers in ⁠${\displaystyle K}$⁠. In a non-Archimedean local field ⁠${\displaystyle K}$⁠, an order is a subring which is generated by finitely many elements of non-negative valuation. In that case, the maximal order, denoted ⁠${\displaystyle {\mathcal {O}}_{K}}$⁠, is the valuation ring formed by all elements of non-negative valuation.

Giving the same name to such seemingly different notions is motivated by the local–global principle that relates properties of a number field with properties of all its local fields.

In mathematics, an algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that is, every element of L is a root of a non-zero polynomial with coefficients in K. A field extension that is not algebraic, is said to be transcendental, and must contain transcendental elements, that is, elements that are not algebraic.

The algebraic extensions of the field ${\displaystyle \mathbb {Q} }$ of the rational numbers are called algebraic number fields and are the main objects of study of algebraic number theory. Another example of a common algebraic extension is the extension ${\displaystyle \mathbb {C} /\mathbb {R} }$ of the real numbers by the complex numbers.