Degree is the answer to a recurring algebraic question: how much information is needed to build an object from a smaller one? A polynomial has a highest power, a [field extension](/page/Field%20Extension) has a dimension over its base field, and a graded algebra remembers which pieces sit in which layer. These are different constructions, but each uses degree to measure algebraic size in a way that survives change of presentation.

The first warning is that visible complexity is not the same as degree. The polynomial $x^4 - 1$ has degree $4$, even though it factors over $\mathbb{Q}$ as $(x^2 - 1)(x^2 + 1)$ and over $\mathbb{C}$ as $(x - 1)(x + 1)(x - i)(x + i)$. The field $\mathbb{Q}(\sqrt[3]{2})$ is generated by one element, but it has degree $3$ over $\mathbb{Q}$. Degree is designed to ignore the accidental form of an expression and record the invariant algebraic size behind it.

[example: Same Formula, Different Algebraic Size] Consider the same element $\sqrt{2}$ with three different base fields: $\mathbb{Q}$, $\mathbb{Q}(\sqrt{2})$, and $\mathbb{R}$. Over $\mathbb{Q}$, it satisfies the polynomial relation \begin{align*} (\sqrt{2})^2-2=2-2=0. \end{align*} Here $\mathbb{Q}[x]$ denotes the ring of polynomials in the variable $x$ with rational coefficients. Thus $x^2-2\in\mathbb{Q}[x]$ vanishes at $\sqrt{2}$. We check that no nonzero linear polynomial in $\mathbb{Q}[x]$ vanishes at $\sqrt{2}$. Let $ax+b\in\mathbb{Q}[x]$ and suppose \begin{align*} a\sqrt{2}+b=0. \end{align*} If $a=0$, then the equation becomes \begin{align*} b=0, \end{align*} so $ax+b$ is the zero polynomial. If $a\ne 0$, then subtracting $b$ and dividing by $a$ gives \begin{align*} \sqrt{2}=-\frac{b}{a}. \end{align*} Since $a,b\in\mathbb{Q}$ and $a\ne 0$, the quotient $-b/a$ lies in $\mathbb{Q}$. It remains to rule out $\sqrt{2}\in\mathbb{Q}$. Suppose $\sqrt{2}=p/q$ with $p,q\in\mathbb{Z}$, $q\ne 0$, and $\gcd(p,q)=1$. Squaring gives \begin{align*} 2=\frac{p^2}{q^2}. \end{align*} Multiplying by $q^2$ gives \begin{align*} p^2=2q^2. \end{align*} Hence $p^2$ is even, so $p$ is even. Write $p=2r$ with $r\in\mathbb{Z}$. Substituting gives \begin{align*} (2r)^2=2q^2. \end{align*} Thus \begin{align*} 4r^2=2q^2. \end{align*} Dividing by $2$ gives \begin{align*} 2r^2=q^2. \end{align*} So $q^2$ is even, hence $q$ is even. This contradicts $\gcd(p,q)=1$, since both $p$ and $q$ are divisible by $2$. Therefore $\sqrt{2}\notin\mathbb{Q}$, so the least possible degree of a nonzero polynomial in $\mathbb{Q}[x]$ vanishing at $\sqrt{2}$ is $2$. Over $\mathbb{Q}$, the element $\sqrt{2}$ therefore has algebraic degree $2$. Over $\mathbb{Q}(\sqrt{2})$, the element $\sqrt{2}$ is already in the base field, so $x-\sqrt{2}$ is a polynomial in $\mathbb{Q}(\sqrt{2})[x]$. Evaluating it at $\sqrt{2}$ gives \begin{align*} \sqrt{2}-\sqrt{2}=0. \end{align*} The polynomial $x-\sqrt{2}$ is monic and has degree $1$. No nonzero polynomial has degree less than $1$ and root $\sqrt{2}$, because a degree-$0$ polynomial is a nonzero constant and cannot evaluate to $0$. Hence the same element has algebraic degree $1$ over $\mathbb{Q}(\sqrt{2})$. The same calculation works over $\mathbb{R}$ because $\sqrt{2}\in\mathbb{R}$. The polynomial $x-\sqrt{2}\in\mathbb{R}[x]$ is monic, linear, and satisfies \begin{align*} \sqrt{2}-\sqrt{2}=0. \end{align*} The same element therefore has algebraic degree $2$ over $\mathbb{Q}$, but algebraic degree $1$ over any chosen base field that already contains $\sqrt{2}$. [/example]

This example captures the central theme of the chapter. Degree is never just a property of an isolated symbol. It is a property of a polynomial relative to its coefficient ring, of an element relative to a base field, or of a homogeneous object relative to a grading. To use degree well, the ambient algebra must be part of the statement.

## Definition

Before separating the polynomial, field-theoretic, and graded meanings, it is useful to name the common pattern. A degree is never just a bare number; it is an assignment made after a class of objects and a base structure have been fixed.

[definition: Degree] Let $B$ be a fixed base structure, let $\mathcal{C}_B$ be a specified class of admissible algebraic objects over $B$, and let $\mathcal{D}$ be either $\{0\} \cup \mathbb{N}$ or a class of cardinal numbers. A degree on $\mathcal{C}_B$ is a map \begin{align*} \deg_B: \mathcal{C}_B &\to \mathcal{D} \end{align*} that assigns to each object $A \in \mathcal{C}_B$ a numerical invariant $\deg_B A$. [/definition]

This is a meta-definition, not a formal universal construction. The precise definition comes from the particular degree function in use: polynomial degree records a highest exponent, extension degree records a vector-space dimension, and graded degree records membership in a homogeneous layer.

### Polynomial Degree

The most elementary version of degree comes from polynomials. It measures the last nonzero coefficient, so it is sensitive to cancellation but insensitive to factorisation. The next definition is needed because later field-theoretic degrees are built from the same idea of measuring the first place where a nonzero leading term occurs.

[definition: Degree of a Polynomial] Let $R$ be a commutative ring. The polynomial degree over $R$ is the map \begin{align*} \deg_R: R[x] \setminus \{0\} &\to \{0\} \cup \mathbb{N}. \end{align*} For a nonzero polynomial \begin{align*} f &= a_0 + a_1x + \cdots + a_nx^n \end{align*} with $a_n \ne 0_R$, \begin{align*} \deg_R f &= n. \end{align*} [/definition]

The zero polynomial has no degree unless a convention is explicitly stated. The exclusion of the zero polynomial is not cosmetic. If $\deg 0$ were treated as an ordinary integer, the usual rules for multiplication would already fail when a product vanished through zero divisors. More broadly, over rings with zero divisors degree additivity can fail even for nonzero products, because the highest-degree term can cancel while lower terms survive. Many contexts adopt $\deg 0 = -\infty$ for bookkeeping, but the ordinary algebraic definition keeps the zero polynomial separate.

The leading coefficient often controls whether degree behaves exactly as expected under division. Over a general ring, a nonunit leading coefficient can disappear after multiplication or fail to support division.

The next distinction is not just terminology: it singles out the polynomials whose top term is fixed by the unit of the coefficient ring. This is the case in which the leading term is protected from many ring-theoretic pathologies and can later serve as the normal form for minimal polynomials and integrality.

[definition: Monic Polynomial] Let $R$ be a commutative ring. A nonzero polynomial $f \in R[x]$ of degree $n$ is monic if the coefficient of $x^n$ is $1_R$. [/definition]

Monic polynomials are important because their leading coefficient remains the image of $1_R$ under ordinary unital base change, so the leading term cannot disappear in the way a nonunit leading coefficient can. Minimal polynomials are also required to be monic. The monic condition is what makes integrality over rings parallel algebraicity over fields.

### Extension Degree

The field-theoretic version of degree asks a different question: not how high a polynomial goes, but how many independent directions a larger field has over a smaller one. The next definition is needed because a field extension is also a [vector space](/page/Vector%20Space), so [linear algebra](/page/Cambridge%20IB%20Linear%20Algebra) supplies a numerical invariant.

[definition: Degree of a Field Extension] Let $k$ be a field. The extension degree over $k$ is the map \begin{align*} \deg_k^{\mathrm{ext}}: \{K / k : K \text{ is a field extension of } k\} \to \{\text{cardinal numbers}\}. \end{align*} The notation $K / k$ means that $K$ is a field containing a specified copy of $k$; it sends such a field extension to $[K : k]$. The value $[K : k]$ is defined by \begin{align*} [K : k] &= \dim_k K. \end{align*} A field extension $K / k$ is finite if $[K : k]$ is finite. [/definition]

The notation $[K : k]$ deliberately resembles the index of a subgroup. In both settings, the number measures how many independent copies of the smaller object are needed to account for the larger one. For fields, the copies are vector-space directions.