# Definition:Degree of Polynomial/Field

## Definition

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$.

Let $\struct {K, +, \times}$ be a subfield of $F$.

Let $x \in F$.

Let $\displaystyle f = \sum_{j \mathop = 0}^n \paren {a_j x^j} = a_0 + a_1 x + \cdots + a_n x^n$ be a polynomial over $K$ in $x$ such that $a_n \ne 0$.

Then the **degree of $f$** is $n$.

The **degree of $f$** can be denoted $\map \deg f$ or $\partial f$.

## Also known as

The **degree** of a polynomial is also referred to by some sources as its **order**.

## Sources

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Polynomial Equations