__1.1) Fundamental theorem of algebra (D'Alembert)__:

Every non-zero, single-variable, degree $n$ polynomial with complex coefficients has, counted with multiplicity, exactly $n$ complex roots.

For more details see here

__1.2) Number field definition (A):__

Let $F$ be a subset of $\mathbb{C}$ which contains at least two elements. We say that $F$ is a **number field** if the following conditions are met:

a) if $z_1$ and $z_2$ are arbitrary numbers from $F$, then the numbers $z_1 + z_2$, $z_1 - z_2$, $z_1 \cdot z_2$ are also in $F$,

b) if $z_1$ and $z_2$ are arbitrary numbers from $F$ and $z_2 \ne 0$, then $z_1 / z_2$ is also in $F$.

__1.3) Number field definition (B):__

Let $F$ be a subset of $\mathbb{C}$ which contains at least two elements. We say that $F$ is a **number field** if the following conditions are met:

a) if $z_1$ and $z_2$ are arbitrary numbers from $F$, then the numbers $z_1 + z_2$, $z_1 - z_2$, $z_1 \cdot z_2$ are also from $F$;

b) if $z \in F$ and $z \ne 0$, then $z^{-1} \in F$

__1.4) Note:__ Not all fields in mathematics are considered number fields (e.g. finite Galois fields are not considered number fields). The number sets $\mathbb{Q}, \mathbb{R}, \mathbb{C}$ (equipped with the usual algebraic operations) are number fields. The number sets $\mathbb{N}, \mathbb{Z}$ are not number fields.

__1.5) Theorem:__ Every number field is a superset of the field of the rational numbers $\mathbb{Q}$.

__1.6) Notations:__

a) $F_{m \times n}$ - the set of all $m \times n$ matrices with elements from the number field $F$

b) $M_{n}(F)$ - the set of all $n \times n$ square matrices with elements from the number field $F$; these are also called square matrices of order $n$

c) If $A = (a_{ij})_{m\times n}$ is a matrix, then by $A^t$ we denote the transposed matrix of the matrix $A$; it is defined as follows: $A^t = (a_{ji})_{n\times m}$

d) $E_n$ or just $E$ is the identity matrix ($E_n$ just emphasizes the fact that the order of the matrix is $n$)

e) $E_{ij}$ is the matrix from $F_{m \times n}$ which has element $1$ at position $(i,j)$ and zero elements at all other positions $(k,l) \ne (i,j)$

__1.7) Operations on matrices:__

a) sum of matrices: $A+B$

b) multiplication of a matrix with a number: $\lambda A$

c) transposed matrix: $A^t$

__1.8) Property:__

Obviously if $A = (a_{ij})_{m\times n}$ is an arbitrary matrix from $F_{m \times n}$ then $$A = \sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} \cdot E_{ij}$$

__1.9) Definition:__

a) The matrices which have a single row (or a single column) are called **row matrices **(or **column matrices**),** **or **n-tuples, **or also **n-dimensional vectors**.

b) The set of all **n-tuples **with elements from the number field $F$ is denoted by $F^n$

c) The **n-tuples **

$e_1 = (1, 0, 0, \dots, 0, 0),\ e_2 = (0, 1, 0, \dots, 0, 0),\ e_3 = (0, 0, 1, \dots, 0, 0),\ \dots\ ,\ $

$e_{n-1} = (0, 0, 0, \dots, 1, 0),\ e_{n} = (0, 0, 0, \dots, 0, 1)$

are called **unit vectors **or **unit ****n-dimensional vectors**.

__1.10) Note:__ If $a = (a_1, a_2, \dots, a_n) \in F^n$ then obviously $a = a_1 e_1 + a_2 e_2 + \dots + a_n e_n$ This expression for $a$ is called a **linear combination**. In this particular case, $a$ is linear combination of the unit vectors $e_1, e_2, \dots, e_n$

__1.11) Properties of the transposed matrix:__

a) $(A^t)^t = A$

b) $(A+B)^t = A^t + B^t$

c) $(\lambda A)^t = \lambda A^t$ (for any $\lambda \in F$)

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