Remarks

• By X⁰ we mean 1.
• We do not distinguish between the polynomials 0X^k and 0. Terms of the form 0X^k are often left out.
• We usually write X^k instead of 1X^k and -X^k instead of (-1)X^k.
• We reduce an expression like X² + (-4)X³ to X² - 4X³.
• Clearly, the inclusions Z Q R imply the inclusions Z[X] Q[X] R[X].
• `X' is not the exclusive name for an indeterminate. Other symbols that we will occasionally use are, for example, Y or Z.
• The words polynomial and monomial are Greek: poly=many, mono=one, nomos=term, name.