Definition:Degree

Disambiguation

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Degree may refer to:

• Polynomial Theory:
  • Degree of a polynomial: the largest natural number $k \in \N$ such that the coefficient of $x^k$ in $P$ is nonzero.
  • Degree of Element of Free Commutative Monoid

• Number Theory:
  • Degree of an algebraic number: the lowest possible degree of a polynomial over $\Q$ of which the algebraic number is a root.
  • Degree of an algebraic number over a general field: the lowest possible degree of a polynomial over a more general field of which the algebraic number is a root.

• Analysis:
  • Degree of a homogeneous function: $f$ is homogeneous of degree $n$ if and only if $\map f {\alpha \mathbf v} = \alpha^n \map f {\mathbf v}$.
  • Degree of a homogeneous real function: $f$ is homogeneous of degree $n$ if and only if $\map f {t x, t y} = t^n \map f {x, y}$.

• Abstract Algebra:
  • Degree of a Homogeneous Element of a gradation
  • Degree of Field Extension: the dimension of a field extension $E/F$ when $E$ is viewed as a vector space over $F$.
  • Degree of an algebraic element: the lowest possible degree of a polynomial of which the algebraic element is a root.
  • Transcendence Degree: the largest cardinality of an algebraically independent subset $A \subseteq L$, where $L / K$ is a extension of a field $K$.

• Graph Theory:
  • Degree of a vertex: as used in graph theory: the number of edges coming together at a particular vertex.

• Geometry:
  • Degree of Arc (Angular Measure): as used in geometry, and so on: $360$ of them make a full circle.

• Analytic Geometry:
  • Degree of Algebraic Curve: the highest degree of the polynomial equations defining it.

• Topology:
  • Degree (Topology):

• Mechanics:
  • Degrees of Freedom: the number of independent variables needed to specify the configuration of an oscillating system.

• Physics:
  • Degrees Celsius: a temperature scale defined between $0 \cels$, the melting point of water, and $100 \cels$, the boiling point of water.
  • Degrees Fahrenheit: a temperature scale defined between $32 \fahr$, the melting point of water, and $212 \fahr$, the boiling point of water.

• Approximation Theory:
  • Degree of Spline: the maximum degree of the polynomials fitted between the knots of a spline function.

• Statistics:
  • Degrees of Freedom: the number of independent units of information in a sample relevant to the estimation of a parameter or calculation of a statistic.