(i) Let $p$ be a prime,. Show that $\wedge^2 (\mathbb{Z}_p \oplus \mathbb{Z}_p) \neq 0$ , where $\mathbb{Z}_p \oplus \mathbb{Z}_p$ si viewed as $\mathbb{Z}$-module ( with $\mathbb{Z}_p $ I mean $\mathbb{Z}/p \mathbb{Z}$ )
(ii) Let $D = \mathbb{Q}/\mathbb{Z} \oplus \mathbb{Q}/\mathbb{Z}$ . Prove that $\wedge^2 D = 0$
What I have done: for the point (ii) $$D \otimes D = (\mathbb{Q}/\mathbb{Z} \oplus \mathbb{Q}/\mathbb{Z}) \otimes (\mathbb{Q}/\mathbb{Z} \oplus \mathbb{Q}/\mathbb{Z}) \cong \bigoplus_{i=1}^{4} (\mathbb{Q}/\mathbb{Z} \otimes \mathbb{Q}/\mathbb{Z}) $$ but $\mathbb{Q}/\mathbb{Z}$ is divisible, and so $\mathbb{Q}/\mathbb{Z} \otimes \mathbb{Q}/\mathbb{Z} = 0$ .
Thus $\wedge^2 D = 0$ because $\wedge^2 D$ is a submodule of $D \otimes D$ .
I'm stuck on point (i) , I should find a bilinear alternating map defined on $ (\mathbb{Z}_p \oplus \mathbb{Z}_p) \times (\mathbb{Z}_p \oplus \mathbb{Z}_p) $ which is not $0$ , but which one ?
Hint: It may be helpful to write $((a,b),(c,d))\in(\mathbb{Z}_p \oplus \mathbb{Z}_p) \oplus (\mathbb{Z}_p \oplus \mathbb{Z}_p)$ as a $2\times 2$ matrix in $\Bbb Z_p$. Can you think of any maps that are bilinear and alternating on the columns (or rows) of a matrix?