{%hackmd @RintarouTW/About %}

# Field Theory

:::info
A field is a **commutative ring with unity** such that each **nonzero element has a multiplicative inverse**.
:::

The most common infinite fields
$$
\cases{
[\mathbb{Q};+,\cdot]\\
[\mathbb{R};+,\cdot]\\
[\mathbb{C};+,\cdot]
}
$$

- Every field is an **integral domain**
$$
F\in Field, \forall a\in F, a\ne 0, \exists a{^-1}\in F, aa^{-1}=a^{-1}=1\\
if\ \exists a,b\in F, a\ne 0, b\ne 0, ab=0\\
a^{-1}ab=a^{-1}0\\
b=0\ (contradiction\ to\ b\ne 0)\\
\therefore F\ has\ no\ zero-divisors\implies F\in \mathfrak{D}(integral domain)
$$
- Every **finite integral domain** is a field.

### Galois Field $GF(p^n)$

- Any finite field F has order $p^n$ for a prime $p$ and a positive integer $n$.
- For any prime $p$ and any positive integer $n$ there is a field of order $p^n$.
- Any two fields of order $p^n$ are isomorphic.

## Polynomial Rings

### Polynomial over a Ring

### Polynomial Addition

### Polynomial Multiplication

### Properties of Polynomial Rings

### Division Property for Polynomials

### The Factor Theorem

### Reducibility over a Field

## Field Extensions

## Power Series

### Power Series Addition
### Power Series Multiplication
### Properties of Power Series

### Polynomial Units
### Power Series Units

###### tags: `math`