Definition:Degree


 * Polynomial Theory:
 * Degree of a polynomial: the largest natural number $k \in \N$ such that the coefficient of $x^k$ in $P$ is nonzero.
 * Definition: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 $n$ $f \left({\alpha \mathbf v}\right) = \alpha^n f \left({\mathbf v}\right)$.
 * Degree of a homogeneous real function: $f$ is homogeneous of degree $n$ $f \left({t x, t y}\right) = t^n f \left({x, y}\right)$.


 * 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, etc: $360$ of them make a full circle.


 * Topology
 * Degree (Topology):


 * 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.