Let $(D, +, \cdot, \delta)$ be a euclidean domain. Prove that:
a) An element $u \in D \setminus \{0\}$ is an unit if, and only if, $\delta(u) = \delta(1)$.
b) If $u$ is an unit, then for each $x \in D \setminus \{0\}$, $\delta(x) = \delta(ux)$.
c) If $x, d \in D \setminus \{0\}$ satisfy $\delta(x) = \delta(xd)$, then $d$ is an unit.
I managed to finish a), but I'm having a hard time with b) and c).
For b), I tried to prove that $\delta(x) \leq \delta(ux)$ and $\delta(ux) \leq \delta(x)$, but I got nowhere (I probably have to use a), but I'm not seeing how to do that). I tried writing $1 = d \cdot t + r$ and proving $\delta(r) < \delta(d)$ cannot happen, but that didn't work out either.
I'd appreciate any help.
I think I've got it:
For b), notice that $\delta(x) \leq \delta(ux)$ is satisfied by definition. On the other hand, $\delta(x) = \delta(u^{-1}ux) \geq \delta(ux)$.
For c): suppose that $\delta(x) = \delta(xd)$ but $d$ is not a unit. Then $x = xdq + r$ and $0 \neq r = (1-dq)x$, therefore $\delta(r) = \delta((1-dq)x) < \delta(x)$, a contradiction.