The field of complex numbers

Find all complex numbers $z$, of modulus 1, for which $z^{2}+(1+i) z$ takes purely imaginary values. Depict the corresponding locus of points in the plane $\mathbb {C}$.

What can be said about the field $\mathbb {R}(\delta )$, which is obtained from $\mathbb {R}$ by adjoining a complex number $\delta$ satisfying the equality $\delta^{4}=-1$?

Let $A, B \in M_{n}(\mathbb {R})$. Relying on Theorem 1, prove that $\overline{\operatorname {det}(A+i B)}=\operatorname {det}(A-i B)$ (the bar denotes conjugation).

Let $A, B \in M_{n}(\mathbb {R})$,

Applying to the real matrix $C$ elementary transformations of the first and second type over the field of complex numbers $\mathbb {C}$, show that

Using Exercises 3 and 4, give an explanation of the following "strange" fact. The homogeneous square linear system

with complex coefficients $d_{k l}=a_{k l}+i b_{k l}$ and unknowns $z_{i}=x_{i}+i y_{i}$ has a nontrivial solution $\left(z_{1}, \ldots , z_{n}\right)$ precisely when $\operatorname {det}\left(d_{k l}\right)=$ $=a+i b=0$ (see the general remarks on this in item 6, § 3, Ch. 4). This condition leads to two equations $a=0, b=0$, relating $2 n^{2}$ real quantities $a_{k l}, b_{k l}$. On the other hand, the system (*) can be represented as a system of $2 n$ linear homogeneous equations with $2 n$ real unknowns $x_{i}, y_{i}$. Now the condition for nontriviality of the solution is written as the vanishing of a single real determinant of size $2 n \times 2 n$, which gives only one equation between $a_{k l}, b_{k l}$. How can these two results be reconciled?

Bearing in mind that automorphisms of the quadratic field $\mathbb {Q}(\sqrt{d})$ must fix the rational numbers, find the automorphisms of this field.

Answer. The identity map and $a+b \sqrt{d} \mapsto a-b \sqrt{d}$.

What is the sum of all $n$-th roots of unity, $n>1$? What can be said about the sum of the primitive 12th roots and the primitive 15th roots of unity?

Show that $\zeta =(2+i) /(2-i)$ is not a root of unity, although $\left|\zeta \right|=1$.

The set $S^{1}=\left\{ e^{i \varphi } \mid \varphi \in \mathbb {R}\right\}$ (the circle of unit radius) forms, with respect to multiplication in $\mathbb {C}$, a subgroup of the group $(\mathbb {C}^{*}, \cdot )$. Every $\mathbb {R}$-linear map $f: \mathbb {C} \rightarrow \mathbb {C}$ is called orthogonal if $\left(f(z) \mid f\left(z^{\prime }\right)\right)=\left(z \mid z^{\prime }\right)$, i.e., if it preserves the lengths of vectors (the distances between points). Prove that a map $f: \mathbb {C} \rightarrow \mathbb {C}$ is orthogonal precisely when $f(z)=c z$ or $f(z)=c \bar{z}$, where $c \in S^{1}$.

Show that

where $\zeta$ is a primitive $n$-th root of unity.