Fields and Field extensions

huh \(\LaTeX\) what? So I am reading through the section on fields and field extensions but I have to admit it is blowing my mind and I have having a hard time following the signicance of what is going on.

\( x>3 \) One example they give is the extension from \(\mathbb{R}\)(real numbers, I don't have latex setup to do fancy letters here). The polynomial P(x) = x^2 +1 has no real root. The root is i which isn't contained in R. We adjoin i to R and we get a new field C(complex) of the form a+bi with a,b contained in R.

One of the parts that got me(although writing this post is helping me think it through) was the concept of adjoining. I originally assumed that it meant you added that item into the set comprising the field but that doesn't seem to be the case. Adjoining does the funky thing the you see normally with complex numbers. If you adjoin item x, then the field had members of the form a+bx.

Another example is the field Q(rationals) which you can adjoin (2)^1/2 or the square root of 2.

The members of the new field are now of the form a + b*(2)^(1/2).

As a side note that looks really ugly I need to figure how to do nice latex and mathematical symbols on here if I want this to keep dumping funky math on this blog.