A little context is in order. I was trying to find counter-examples to the following statement:
$$\phi : X\rightarrow Y \;\text{injective} \Rightarrow \phi\otimes K : X\otimes_k K \rightarrow Y\otimes_k K \;\text{injective}$$
where $X$ and $Y$ are schemes over a field $k$ and $K/k$ is a field extension. Here, when I say injective I mean on the topological spaces $|X|$ and $|Y|$. Now I know this isn't true since we have the clear counter-example: $X=\text{Spec}(\mathbb{C})$, $Y=\text{Spec}(\mathbb{R})$, $k=\mathbb{R}$ and $K=\mathbb{C}$.
Now I want to clearify that I just started studying scheme theory and one of my teachers sent me another counter-example with a more "geometric" feel to it. We take $k=\mathbb{Q}$ the affine line $\mathbb{A}^1 _{\mathbb{Q}}$ and we look at the algebra morphism given by
$$\mathbb{Q}[t]\rightarrow \mathbb{Q}[t],\quad t\mapsto t^n$$
where $n$ is odd and different from 1. Now the fact which isn't clear for me is injectivity, my teacher says it comes from the following fact:
Let $a$ and $b$ be two irreducible monic polynomials in $\mathbb{Q}[t]$, then $$\forall u\in \mathbb{Q}[t],\{a|u(t^n)\iff b|u(t^n)\}\Rightarrow a=b$$
I have no clue how to show this fact. If this is true however, we can consider $\zeta_n$ a $n$ primitive root of unity and take $K=\mathbb{Q}(\zeta_n)$.
Thank you for any clues or answers.