Proposition 13

— Let $\mathscr{I}$ be the set of irreducible monic polynomials in $K[X]$. Let f be a non-zero element of $K[X]$ and a its leading coefficient; then there exists precisely one family of positive integers $(v_p)_{p \in \mathscr{S}}$ with finite support, such that we have a decomposition

It suffices to prove the proposition when $f$ is monic, that is when $\alpha = 1$. We shall argue by induction on the degree $n$ off, the case $n = 0$ being trivial. Suppose then that $n \geq 1$ and that the proposition has been established for all polynomials of degree $< n$.

Let $E$ be the set of monic polynomials $\neq 1$ which divide $f$; we have $f \in E$ hence $E$ is not empty and there exists in $E$ a polynomial $g$ of least degree. It is clear that $g$ is irreducible and there exists a monic polynomial $h$ of degree $< n$ such that $f = gh$; by the induction hypothesis $h$ is the product of a finite family of irreducible monic polynomials, hence $f$ has the same property. This proves the existence of the decomposition (18).

Let us now prove the uniqueness of the decomposition (18). Let $(w_p)_{p \in \mathscr{S}}$ be a family of positive integers with finite support, such that $f = \prod_{p \in \mathscr{S}} p^{w_p}$. Since $f$ is of degree $n \geq 1$, there exists $p \in \mathscr{S}$ such that $w_p > 0$; if we had $v_p = 0$, $f$ would be the product of a family of elements of $\mathscr{S}$ distinct from $p$, hence it would be prime to $p$ (IV, p. 13, Cor. 6), contrary to the fact that $p$ divides $f$. By the induction hypothesis the polynomial $f/p$ admits a unique decomposition of type (18); hence we conclude the equality $w_q = v_q$ for every $q \in \mathbf{4}$.

Let $f$ be a non-zero polynomial in $\mathbf{K}[X]$. We shall say that $f$ has no multiple factors if the exponents $v_p$ in the decomposition (18) are all $\leq 1$; it comes to the same to say that $f$ is the product of a finite sequence of pairwise distinct irreducible polynomials, or also that $f$ is not divisible by the square of any non-constant polynomial of $\mathbf{K}[X]$.