Consider $R=k[x_1,x_2,...x_n]$. Difference between $(f_1,..,f_k)$ and $k[f_1,...f_k]$

Question

I'm confused on the difference between these two objects. My book says that we can see the difference when considering the simple case where $f_1=x^2\in k[x]$, and see that $1\in k[x^2]$, but $1\notin (x^2)$.

However I couldn't figure out why $1\in k[x^2]$. Could anyone help illustrate this?

Answer 1

$k[x^2]$ denotes the set of polynomials in $x^2$ with coefficients in the field $k$, i.e. the set of (ordinary) polynomials with only monomials of even degree (+0). $1$ has degree $0$, as all non-zero constants, and if I remember well, $0$ is even. So $1$ belongs to $k[x^2]$.