# Definition:Degree

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## 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.
- 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 $n$**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 $\map f {t x, t y} = t^n \map f {x, y}$.

- Degree of a homogeneous function: $f$ is

- 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.

- 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.