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.



Given a unital ring kk, a monic polynomial over kk is a polynomial with coefficients in kk, whose highest order coefficient is 11.

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

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

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


Last revised on December 10, 2022 at 14:03:02. See the history of this page for a list of all contributions to it.