Problem 1 (DF, Problem 7, p. 455)

Let $R=\mathbb{Z}[x,y]$ and let $I=(x,y)$. Define a map

$$\varphi (ax+by,a^{\prime }x+b^{\prime }y)=ad-bc(\mathrm{mod}I).$$

Prove that this is a well defined alternating map from $I\times I\to \mathbb{Z}$.

Problem 2 (DF, Problem 12, p. 455)

Let $F$ be of characteristic $2$ and let $V$ be a vector space over $F$. Prove that an alternating bilinear map on $F$ is symmetric, but that not every symmetric bilinear map is alternating.

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