Exercise 111.1.4. Let $k$ be a field. Show that the following pairs of $k$-algebras are not isomorphic:

$k[x_1, \ldots , x_ n]$ and $k[x_1, \ldots , x_{n + 1}]$ for any $n\geq 1$.

$k[a, b, c, d, e, f]/(ab + cd + ef)$ and $k[x_1, \ldots , x_ n]$ for $n = 5$.

$k[a, b, c, d, e, f]/(ab + cd + ef)$ and $k[x_1, \ldots , x_ n]$ for $n = 6$.

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