Complex conjugation and its properties

The number $z + \overline{z}$ is real, $z - \overline{z}$ is purely imaginary, $z\overline{z}$ is real and not negative.

Let $z = z_{0}$ be a root of a polynomial $P(z)$ with real coefficients. Prove that $P\left(\bar{z}_{0}\right) = 0$, i.e., $\bar{z}_{0}$ is also a root of the polynomial $P(z)$.

Suppose $w, z \in \mathbb {C}$. Verify the following equalities and inequalities.

Suppose $p \in \mathcal{P}(\mathbb {C})$. Define $q: \mathbb {C} \rightarrow \mathbb {C}$ by

Prove that $q$ is a polynomial with real coefficients.

Prove that:

Prove that:

Prove that if from given complex numbers $z_{1}, z_{2}, \ldots , z_{n}$, by applying finitely many operations of addition, subtraction, multiplication and division, a number $z$ is obtained, then from the numbers $\bar{z}_{1}, \bar{z}_{2}, \ldots , \bar{z}_{n}$, by applying the same operations, the number $\bar{z}$ is obtained.

Prove that the determinant

where $z_{1}, z_{2}, z_{3}$ are complex numbers and $a, b, c$ are real numbers, is a purely imaginary number.

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).