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Degree of a field extension, the Glossary

In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension.^[1]

30 relations: Abstract algebra, Algebraic curve, Associative property, Basis (linear algebra), Cardinal number, Cardinality, Cartesian product, Complex number, Dimension (vector space), Distributive property, Division ring, Field (mathematics), Field extension, Finite field, Function field of an algebraic variety, Galois theory, Group theory, Index of a subgroup, Lagrange's theorem (group theory), Linear independence, Linear span, Mathematics, Number theory, Prime number, Rational function, Rational number, Real number, Tower of fields, Transcendental extension, Vector space.
Field extensions

Abstract algebra

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements.

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Algebraic curve

In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables.

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Associative property

In mathematics, the associative property is a property of some binary operations that means that rearranging the parentheses in an expression will not change the result.

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Basis (linear algebra)

In mathematics, a set of vectors in a vector space is called a basis (bases) if every element of may be written in a unique way as a finite linear combination of elements of.

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Cardinal number

In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set.

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Cardinality

In mathematics, the cardinality of a set is a measure of the number of elements of the set.

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Cartesian product

In mathematics, specifically set theory, the Cartesian product of two sets and, denoted, is the set of all ordered pairs where is in and is in.

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Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.

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Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field.

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Distributive property

In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality x \cdot (y + z).

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Division ring

In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined.

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Field (mathematics)

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.

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In mathematics, particularly in algebra, a field extension (denoted L/K) is a pair of fields K \subseteq L, such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Degree of a field extension and field extension are field extensions.

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Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

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Function field of an algebraic variety

In algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.

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Galois theory

In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory.

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Group theory

In abstract algebra, group theory studies the algebraic structures known as groups.

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Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G. The index is denoted |G:H| or or (G:H).

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Lagrange's theorem (group theory)

In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group, the order (number of elements) of every subgroup of divides the order of.

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Linear independence

In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector.

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Linear span

In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted, pp.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Number theory

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions.

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Prime number

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers.

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Rational function

In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials.

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Rational number

In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

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Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

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Tower of fields

In mathematics, a tower of fields is a sequence of field extensions The name comes from such sequences often being written in the form A tower of fields may be finite or infinite. Degree of a field extension and tower of fields are field extensions.

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Transcendental extension

In mathematics, a transcendental extension L/K is a field extension such that there exists an element in the field L that is transcendental over the field K; that is, an element that is not a root of any univariate polynomial with coefficients in K. In other words, a transcendental extension is a field extension that is not algebraic.

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Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''.

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References

[1] https://en.wikipedia.org/wiki/Degree_of_a_field_extension

Also known as Degree (field extension), Degree of an extension, Degree theorem, Extension field degree, Extension field index, Finite extension, Finite field extension, Tower Law.