ENT - Brief Refresher to Field Extensions

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Introduction

This is a very brief refresher of Field extensions. We assume basic knowledge of fields.

Field Extensions

Given the field of integers $\mathbb{Z} = \{…, -3, -2,-1, 0, 1 ,2,3,4,…\}$ , lets say I want to make this field larger by adding an extra element $\sqrt{2}$ to this field.

$K$ = $\{…, -3, -2,-1, 0, {\bf\sqrt{2}},1 ,2,3,4,…\}$

We would no longer have a field, because we no longer have closure under addition or multiplication; If I take 2 elements $\sqrt{2}$ and $5$ from $K$. The result $\sqrt{2} + 5$ is not in $K$. Lets add it!

$K$ = $\{…, -3, -2,-1, 0, {\bf\sqrt{2}}, {\bf\sqrt{2} + 5},1 ,2,3,4,…\}$

But then we can argue that ${\bf\sqrt{2} + 3}$.

No worries, there is a way! We must simply add all elements of the form $a + b\sqrt{2}$ where $a,b \in \mathbb{Z}$.

So we can define $\mathbb{Z[\sqrt{2}]}$ as ${a + b\sqrt{2} : a,b \in \mathbb{Z}}$.

We call $\mathbb{Z[\sqrt{2}]}$ an extension of $\mathbb{Z}$. Notice that when $b=0$ we simply have our original field.

Field elements - reducibility.

Lets look at the $\mathbb{Z[\sqrt{4}]}$ . $\mathbb{Z[\sqrt{4}]}$ = $\{…, -3, -2,-1, 0, {\bf\sqrt{4}}, 1 ,2,3,4,…}$ = $\{…, -3, -2,-1, 0, {\bf2}, 1 ,2,3,4,…\}$ = $\{…, -3, -2,-1, 0, 1 ,2,3,4,…\}$ = $\mathbb{Z}$

The last simplification is because when sets with duplicate elements are the same.

Why did it not add more elements?

As observed, if the element you are adding to the field it already in the field, then you are not extending it.

It worked for $\sqrt{2}$ but not $\sqrt{4}$ because $\sqrt{4}$ can be reduced further to $2$ which is already in $\mathbb{Z}$.

Observe $\sqrt{8} = 2\sqrt{2} \in \mathbb{Z[\sqrt{2}]}$ . so using $\sqrt{8}$ does not extend further than $\mathbb{Z[\sqrt{2}]}$

Square-free

We note that if d is square-free, then $\mathbb{Z[\sqrt{d}]}$ will extend $\mathbb{Z}$.

First we must find an element that is not in $\mathbb{R}$ and cannot be reduced further. We note that the square root of a negative number is not defined in set of real numbers.

Extending the real number : $\mathbb{R[\sqrt{-1}]} = {a+b\sqrt{-1} : a,b \in \mathbb{R}}$

We call $\mathbb{R[\sqrt{-1}]}$ the set of complex numbers and denote it with the symbol $\mathbb{C}$. The complex numbers can also be extended, but the element that extends it will not be the root of a polynomial. This follows from the fundamental theorem of algebra; where every polynomial of degree-n has n complex roots.