Let $k$ be a field, and let $s_1, ..., s_n$ be the elementary symmetric polynomials in $k[X_1, ..., X_n]$. The fundamental theorem of symmetric polynomials tells us that
Each symmetric polynomial in $k[X_1, ..., X_n]$ can be written as a polynomial of $s_1, ..., s_n$. Moreover, the polynomials $s_1, ..., s_n$ are algebraic independent.
Hence, if we define $k[X_1, ..., X_n]^{S_n}$ to be the set of symmetric polynomials in $k[X_1, ..., X_n]$, then we have an algebra isomorphism $f \colon k[X_1, ..., X_n]^{S_n} \to k[t_1, ..., t_n]$ that is determined by $f(s_i)=t_i$.
My question is, $k[t_1, ..., t_n]$ is basically the same thing as $k[X_1, ..., X_n]$. By the preceding argument, the we will have $k[X_1, ..., X_n]=k[X_1, ..., X_n]^{S_n}$, which is weird. Where did I make a mistake?