Nice exercises on Hilbert's basis theorem

A) Obligatory exercise, to be committed to memory.
Any finitely generated algebra over a noetherian ring is noetherian.

B) If $A$ is a PID and $f\in A$ , then the fraction ring $A_f=S^{-1}A$ with $S=\{1,f,f^2,f^3,\cdots\}$ is noetherian.
Amusing example: the ring of all decimal numbers, i.e. those that can be written with finitely many digits after the decimal point (like $-3.1415926535$), is noetherian.

C) The image of $f:\mathbb R\to \mathbb R^n: t\mapsto (t,t^2,\cdots, t^n)$ is an algebraic variety.

D) If $k$ is a field, then any $k$-subalgebra of the $k$-algebra $k[X]$ is noetherian.

E) Is a noetherian algebra over a field finitely generated over that field?

F) a) Is the $\mathbb Q$-algebra $\mathbb Q[1+\sqrt 2,1+2\sqrt 2,\cdots , 1+n\sqrt 2,\cdots ]$ finitely generated over $\mathbb Q$ ?

$\; \; \:$ b) Is the polynomial ring in infinitely many variables $\mathbb Q[X_1,X_2,\cdots , X_n,\cdots ]$ finitely generated over $\mathbb Q$?
$\; \; \:$ c) Is the ring $\mathbb Q(X)$ noetherian? Is it a finitely generated algebra over $\mathbb Q$ ?

NB These are not necessarily corollaries of Hilbert's theorem but are definitely very related to it.

Solution of D)
Let $A\subsetneq k[X]$ be a $k$-subalgebra and let $f(X)=X^n+q_1X^{n-1}+\cdots+q_n\in A \quad (n\geq 1)$.
Then the monic polynomial $T^n+q_1T^{n-1}+\cdots+q_n-f(X)\in (k[f(X)])[T]$ kills $X$, so that $k[X]$ is integral over $k[f(X)]$ and thus finite over $k[f(X)]$ ( Atiyah-Macdonald,Remark page 60).
And now for the killing: $k[f(X)]$ is a noetherian ring by A) and $k[X]$ is a noetherian module over $k[f(X)]$ by finiteness (Atiyah-Macdonald, Proposition 6.5).
Then the $k[f(X)]$-submodule $A$ of $k[X]$ is finitely generated over $k[f(X)]$ as a module and a fortiori as an algebra over the noetherian ring $k[f(X)]$.
Reapplying A), we see that $A$ is a noetherian ring. Et voilà!

Edit: Warning!
The assertion D) becomes false if the field $k$ is replaced by a ring, even a nice noetherian one like $\mathbb Z$:
For example the $\mathbb Z$-subalgebra (which just means the subring!) $A\subset \mathbb Z[X]$ defined by $$A=\mathbb Z\oplus 2\mathbb Z \cdot X\oplus 2\mathbb Z \cdot X^2\oplus 2\mathbb Z \cdot X^3\oplus\cdots =\mathbb Z\oplus (\bigoplus_{i\geq 1} 2\mathbb Z \cdot X^i)$$ is not noetherian since its ideal $$ \langle 2X,2X^2,2X^3,\cdots\rangle \subset A$$ is not finitely generated .

Here's an exercise from Ravi Vakil. The game of Chomp starts with a half-infinite checkboard with squares labeled $(n, m)$ for integers $n, m\geq 0$, with a cookie on each square. Two players take turns choosing a cookie (i.e., not an empty square) and eating all the cookies on that square and the ones that are not to the left or below it; that is, if a player chooses the cookie on $(n, m)$, then they eat all the cookies on $(n', m')$ for $n' \geq n$ and $m'\geq m$. The cookie on $(0, 0)$ is poisoned, and the player who eats it immediately loses. Prove that any game of Chomp ends after finitely many moves.

For the connection with the Hilbert basis theorem, consider $(n, m)$ as representing the monomial $X^n Y^m\in R[X, Y]$. More generally, consider the analogous game on $(\mathbb{Z}^{\geq 0})^d$.

Not an application but a variation on the proof technique: show that if $A$ is a noetherian commutative ring, then so is the ring of formal power series $A[[t]]$.

If you are teaching algebraic geometry, then one should mention that every zero set (in $n$-dimensional affine space over a field) is the union of finitely many irreducible zero sets, i.e. zero sets coming from prime ideals. This is essentially the geometric translation of the statement that in a noetherian ring, every radical ideal is the intersection of finitely many prime ideals.