Unless otherwise stated, in this section we denote by K an arbitrary field and by K[X] the ring of polynomials in one indeterminate X and with coefficients from K. Since K[X] is a PID, it enjoys many properties similar to those of . To start with, we take a look at these properties. Then we introduce the concept of algebraic elements and discuss how irreducible polynomials can be used to construct (algebraic) extensions of fields. When no confusions are likely, we denote a polynomial by f only.
2.6.1. Elementary Properties