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Degree (mathematics)

In mathematics, there are several meanings of degree depending on the subject.

Unit of angle

A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of plane angle, representing 1360 of a full rotation. When that angle is with respect to a reference meridian, it indicates a location along a great circle of a sphere, such as Earth (see Geographic coordinate system), Mars, or the celestial sphere.

Degree of a polynomial

The degree of a term of a polynomial in one variable is the exponent on the variable in that term; the degree of a polynomial is the highest such degree. For example, in 2x3 + 4x2 + x + 7, the term of highest degree is 2x3; this term, and therefore the entire polynomial, are said to have degree 3.

For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree of the polynomial is again the highest such degree. For example, the polynomial x2y2 + 3x3 + 4y has degree 4, the same degree as the term x2y2.

Degree of an algebraic number

The degree of an algebraic number is the smallest degree of a non-trivial polynomial in one variable with rational coefficients having said algebraic number as a root. For instance, any rational number $q$ is degree 1 since it is the root of the polynomial $xmapsto x-q$.

Additionally, the square root of any non-square positive integer, say $sqrt n$, is degree 2, as it is the root of $xmapsto x^2-n$.

Degree of a field extension

Given a field extension K/F, the field K can be considered as a vector space over the field F. The dimension of this vector space is the degree of the extension and is denoted by [K : F].

Degree of a vertex in a graph

In graph theory, the degree of a vertex in a graph is the number of edges incident to that vertex — in other words, the number of lines coming out of the point. In a directed graph, the indegree and outdegree count the number of directed edges coming into and out of a vertex respectively.

Topological degree

In topology the term degree is used for various generalizations of the winding number in complex analysis. See topological degree theory.

Degree of freedom

A degree of freedom is a concept in mathematics, statistics, physics and engineering. See degrees of freedom.

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