01 - Fields

Field

Set $\mathbf{F}$ with two binary operations $+$ and $\times$ is a field if

$$(\mathbf{F},+,0)\text{ and }(\mathbf{F}\setminus \{0\},\times ,1)\text{ are abelian groups}$$

with the distributive law holding

$$(a+b)c=ac+bc\text{ for all }a,b,c\in \mathbf{F}$$

Characteristic of a field $\mathbf{F}$

Smallest integer $p$ such that

$$1+1+\cdots +1(p\text{ times})=0$$

where $p$ is known as the characteristic of $\mathbf{F}$
Otherwise if $p$ doesn’t exist then the characteristic of $F$ is defined to be $0$

Note that $p$ is always prime

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Algebraically Closed field

Let $\mathbf{F}$ be a field then

$\mathbf{F}$ is algebraically closed if every non-constant polynomial in $\mathbf{F}[x]$ has a root in $\mathbf{F}$

Algebraic Closure of $\mathbf{F}$

Let $\overline{\mathbf{F}}$ be a algebraically closed field containing $\mathbf{F}$ such that
There does not exist a smaller algebraically closed field $L$ with

$$\overline{\mathbf{F}}\subseteq L\subseteq \mathbf{F}$$

Then $\overline{F}$ is called the algebraic closure of $\mathbf{F}$

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Special Types of Matrix Groups

Let

$$\begin{array}{rlrl}O(n)& =\{A\in M_{n\times n}(\mathbb{R})\mid A^{T}A=\mathrm{Id}\}& & \text{orthogonal group}\\ SO(n)& =\{A\in O(n)\mid \det A=1\}& & \text{special orthogonal group}\\ U(n)& =\{A\in M_{n\times n}(\mathbb{C})\mid {\overline{A}}^{T}A=\mathrm{Id}\}& & \text{ unitary group}\\ SU(n)& =\{A\in U(n)\mid \det A=1\}& & \text{ special unitary group}\end{array}$$