How do I argue that $k (X)/k$ is not an algebraic extension? I think since $X$ is indeterminant there exists no non-zero polynomial in $k [X]$ having $X$ as a root.Hence $X$ is transcendental over $k$.Therefore the extension $k [X]/k$ is not algebraic rather it is transcendental.
But I am not completely convinced with my argument.Are there more intuitive way to view this fact? If such things do exist please let me know somebody. Then it will be really helpful for me.
Thank you very much.