Math and stuff

Proposition Compute $\sqrt{i}$, $\sqrt{-i}$, $\sqrt{1 + i}$, $\sqrt{\frac{1 - i\sqrt{3}}{2}}$. Solution 1 Let $a + bi = \sqrt{i}$. By squaring both sides, we obtain $a^2 = b^2$ and $2ab = 1$. If $a = b$, then $a = \pm 1/\sqrt{2}$. It is impossible that $a + b = 0$ because...

Proposition Find the values of $(1 + 2i)^3$. $\frac{5}{-3 + 4i}$. $\big(\frac{2 + i}{3 - 2i}\big)^2$. $(1 + i)^n + (1 - i)^n$. Solution $(1 + 2i)^3 = 1 + 3 \cdot 2i + 3 \cdot (2i)^2 + (2i)^3 = -11 - 2i$. $5 / (-3 + 4i) = 5(3...

Proposition Let $e_1, \cdots, e_n$ be the usual basis of $\mathbb{R}^n$ and let $\phi_1, \cdots, \phi_n$ be the dual basis. Show that $\phi_{i_1} \wedge \cdots \wedge \phi_{i_k}(e_{i_1}, \cdots, e_{i_k}) = 1$. What would the right side be if the factor $(k + l)!/k!/l!$ did not appear in the definition of...

Proposition Let $S$ be a multiplicatively closed subset of a ring $A$, and let $M$ be a finitely generated $A$-module. Prove that $S^{-1}M = 0$ if and only if there exists $s \in S$ such that $sM = 0$. Solution Suppose that there exists $t \in S$ such that $tM...

Proposition Let $A$ be a ring, $a$ an ideal, $M$ an $A$-module. Show that $(A / a) \otimes M$ is isomorphic to $M / aM$. Solution $a \rightarrow A \rightarrow A / a \rightarrow 0$ is an exact sequence with the inclusion map $i: a \rightarrow A$ and the canonical...