Field Extension - Simple example

 Real numbers for a Field, so do complex numbers and rationals. What is deeply interesting is that we can construct several other fields based on an existing field. These are called Field extensions.

Standard example of such Field extension goes like this. Let $Q$ be a rational field. Consider polynomial $X^2 -2 \in Q$. Clearly roots of the this polynomial $X=\sqrt{2}$ and $X=-\sqrt{2}$ don't belong to $Q$ as there are irrationals.

One nice things we can do is to extend $Q$ by adding these roots to form a new field. Let's call it $Q(\sqrt{2})$. Now $Q(\sqrt{2})$ consists of roots of polynomials in $Q[X]$ whose roots involve $\sqrt{2}$. 

For example, a polynomial $X^2−2 X−17$. If you solve this quadratic, you will notice that the roots are $1+3\sqrt{2}$ and $1-3\sqrt{2}$. Obviously we want to accommodate all such roots in our new extended field $Q(\sqrt{2})$.

Thus we are forced to consider $a+b\sqrt{2}$ as a general element of this extended field with elements ${1,\sqrt{2}}$ as generating set.

Now, this new field $Q(\sqrt{2})$ forms a vector space over $Q$.

Elements: We can add any two elements and resulting element is still in $Q(\sqrt{2})$.

Scalar multiplication:  Multiplying an element by a scalar form elements that belong to $Q(\sqrt{2})$.

Zero vector: Additive identity works like zero vector.

This setup allows for the exploration of linear algebra concepts within field extensions, including the basis, dimension, and linear transformations, offering a rich framework for understanding more complex algebraic structures.