Field extension in the context of "Number field"

In mathematics, the hyperreal numbers, denoted ${\displaystyle \ast \mathbb {R} }$, are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number ${\displaystyle x}$ is said to be finite when ${\displaystyle |x|<n}$ for some integer ${\displaystyle n}$. Similarly, ${\displaystyle x}$ is said to be infinitesimal when ${\displaystyle |x|<1/n}$ for all positive integers ${\displaystyle n}$. The term "hyper-real" was introduced by Edwin Hewitt in 1948.

The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about ${\displaystyle \mathbb {R} }$ are also valid in ${\displaystyle *\mathbb {R} }$. For example, the commutative law of addition, ${\displaystyle x+y=y+x}$, holds for the hyperreals just as it does for the reals; since ${\displaystyle \mathbb {R} }$ is a real closed field, so is ${\displaystyle *\mathbb {R} }$. Similarly, since ${\displaystyle \sin({\pi n})=0}$ for all integers ${\displaystyle n}$, one also has ${\displaystyle \sin({\pi H})=0}$ for all hyperintegers ${\displaystyle H}$. The transfer principle for ultrapowers is a consequence of Łoś's theorem of 1955.

In mathematics, a Galois module is a G-module, with G being the Galois group (named for Évariste Galois) of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.

A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations f₁ = 0, ..., f_h = 0 where the f_i are polynomials in several variables, say x₁, ..., x_n, over some field k.

A solution of a polynomial system is a set of values for the x_is which belong to some algebraically closed field extension K of k, and make all equations true. When k is the field of rational numbers, K is generally assumed to be the field of complex numbers, because each solution belongs to a field extension of k, which is isomorphic to a subfield of the complex numbers.

In mathematics, an algebraic function field (often abbreviated as function field) of ${\displaystyle n}$ variables over a field ${\displaystyle k}$ is a finitely generated field extension ${\displaystyle K/k}$ which has transcendence degree ${\displaystyle n}$ over ${\displaystyle k}$. Equivalently, an algebraic function field of ${\displaystyle n}$ variables over ${\displaystyle k}$ may be defined as a finite field extension of the field ${\displaystyle K=k(x_{1},\dots ,x_{n})}$ of rational functions in ${\displaystyle n}$ variables over ${\displaystyle k}$.

In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.

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.

In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory—indeed in any area where fields appear prominently.