"Suppose k = Q(√d)$k = \mathbb{Q}(\sqrt{d})$ is a real quadratic field (d > 1$d > 1$ a square-free integer) with fundamental unit ε unit $\varepsilon$, normalized as usual so that ε > 1$\varepsilon > 1$ with respect to the embedding of k$k$ into R$\mathbb{R}$ for which √d > 0 which $\sqrt{d} > 0$. If Norm_{k/Q}(ε) = +1 If $\operatorname{Norm}_{k/\mathbb{Q}}(\varepsilon) = +1$, then, by Hilbert’s Theorem 90, ε = σ(α)/α (1)$$ \tag{1}\label{Hilbert90} \varepsilon = \sigma(\alpha)/\alpha $$ for some α ∈ Q(√d)$\alpha \in \mathbb{Q}(\sqrt{d})$, which may be assumed to be an algebraic integer (for example, take α = σ(ε) + 1 $\alpha = \sigma(\varepsilon) + 1$). The principal ideal (α)$(\alpha)$ is invariant under σ $\sigma$ (that is, is an ambiguous ideal) since σ(α)$\sigma(\alpha)$ differs from α$\alpha$ by a unit, so the element α satisfying (1)$\alpha$ satisfying \eqref{Hilbert90} can be further chosen so that the ideal (α)$(\alpha)$ is the product of distinct ramified primes."
I got this from some paper.
Why (α)$(\alpha)$ can be written as the product of distinct ramified primes?
