# Wednesday, September 02

Recommended exercises:

- 0.9
- 0.5 (easy)
- 0.6a
- 0.10

Taken:

- 0.11
- 0.3
- 0.4


:::{.exercise title="0.5"}
Let $R_1, R_2$ be two $k\dash$algebras that are also domains with fraction fields $K_i$.

Show that $R_1 \tensor_k R_2$ is a domain $\iff$ $K_1 \tensor_k K_2$ is a domain.
:::



:::{.exercise title="0.9"}
Let $k$ be a field and $d\geq 2$ with $4\notdivides d$ and $p\in k[x]$ a polynomial of positive degree.

Factor $p$ in $\bar k[x]$ as $\prod_{i=1}^r (x-a_i)^{e_i}$, and suppose there is some $i$ such that $d\notdivides e_i$.
Show that
\[  
f(x,y) \da y^d - p(x)  \in k[x, y]
\]
is geometrically irreducible.

Conclude that
\[  
ff\qty{ k[x, y] / \gens{f} }
.\]
is a regular one-variable function field over $k$.

:::

:::{.solution}
Recall:

- For $L/K$, 
- A polynomial $f \in k[t_i]$ is *geometrically irreducible* iff $f \in \bar k[t_i]$ is irreducible as a polynomial, i.e. if $f =pq \implies p=1$ or $q=1$.
- A field extension $L/k$ is *regular* iff any of the following conditions hold:
  - $\kappa(k) = k$ and $L/k$ is separable, where $\kappa(k)$ is the field of elements of $L$ algebraic over $k$
  - $L\tensor_k \bar k$ is a domain or a field.
  - For all $L'/k$, $L\tensor_k L'$ is a domain.
:::