nLab monic polynomial

This entry is about the basic notion of “monic” in relation to polynomials in commutative algebra. For the notion of “monic” in relation to morphisms in category theory, see at monomorphism.

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Definition

Given a unital ring $k$, a monic polynomial over $k$ is a polynomial with coefficients in $k$, whose highest order coefficient is $1$.

A root of a monic polynomial over $k$ is by definition an algebraic integer over $k$.

Here algebraic integer usually means algebraic integer over $\mathbf{Z}$. All algebraic integers form a field called the integral closure of $\mathbf{Z}$ in $\mathbf{C}$.

On the other hand, for a number field $K$, an integer in $K$ is an algebraic integer over $\mathbf{Z}$ which is in $K$; all integers in $K$ form a ring of integers $\mathcal{O}_K$ of the number field $K$.

References

• Wikipedia, Algebraic number