I've been solving some problems from my Galois Theory course and I want to check if the solution I came up with is correct. The question was:
Given that an element $t$ is transcendent over a field $K$, is $t+1$ also transcendent over $K$?
What I did is notice that, since $1\in K(t+1)$ and $(t+1)-1=t\in K(t+1)$, then I know $K(t)\subseteq K(t+1)$, and then $[K(t):K]$ divides $[K(t+1):K]$. Then I conclude that $[K(t+1):K]=\infty$ because $[K(t):K]=\infty$ (because $t$ is transcendent over $K$). Lastly, since $K(t+1)$ is simple and infinite over $K$, I can conclude it's transcendent, hence $t+1$ is transcendent over $K$.
Is my proof correct? If not, where did I go wrong? Any help will be appreciated, thanks in advance.
It looks correct, but it is more simple to say that, if $t+1$ is a root of $P(x)\in K[x]$, then $t$ is a root of $P(x+1)$.