Exercise 3, § 1

a) In the algebra $A[X]$ of polynomials in one indeterminate over A, the mapping $(u, v) \mapsto u(v)$ is an internal law of composition; show that this law is associative and distributive on the left with respect to addition and multiplication in $A[X]$. b) If $A$ is an integral domain show that the relation $u(v) = 0$ implies that $u = 0$ or $v$ is constant, and that when $u \neq 0$ and $\deg v > 0$, the degree of $u(v)$ is equal to the product of the degrees of $u$ and $v$. c) From now on assume that $A$ is a field. Let u and v be two polynomials of degree $> 0$ in $A[X]$ and f a polynomial of degree $> 0$; show that if q is the quotient and r the remainder in the Euclidean division of u by v, $q(f)$ and $r(f)$ are the quotient and remainder in the Euclidean division of $u(f)$ by $v(f)$. d) For each polynomial f of $A[X]$ denote by I(f) the set of all polynomials of the form $u(f)$, where u ranges over $A[X]$; this is a subring of $A[X]$. For $I(f) = I(g)$ it is necessary and sufficient that $g = \lambda f + \mu$, where $\lambda \neq 0$ and $\mu$ are in $A$. e) Let f and g be two polynomials of degree $> 0$ in $A[X]$; show that $I(f) \cap I(g)$ is either equal to A or of the form $I(h)$, where h is a polynomial of degree $> 0$ (excluding the first possibility, consider in $I(f) \cap I(g)$ a polynomial h of least possible degree $> 0$).