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Important: to view this notebook properly you will need to execute the cell above, which assumes you have an Internet connection. It should already be selected, or place your cursor anywhere above to select. Then press the "Run" button in the menu bar above (the right-pointing arrowhead), or press Shift-Enter on your keyboard.

ParseError: KaTeX parse error: \newcommand{\lt} attempting to redefine \lt; use \renewcommand

It is natural to ask whether or not some field FF is contained in a larger field. We think of the rational numbers, which reside inside the real numbers, while in turn, the real numbers live inside the complex numbers. We can also study the fields between Q{\mathbb Q} and R{\mathbb R} and inquire as to the nature of these fields.

More specifically if we are given a field FF and a polynomial p(x)∈F[x],p(x) \in F[x]\text{,} we can ask whether or not we can find a field EE containing FF such that p(x)p(x) factors into linear factors over E[x].E[x]\text{.} For example, if we consider the polynomial

p(x)=x4−5x2+6\begin{equation*} p(x) = x^4 -5 x^2 + 6 \end{equation*}

in Q[x],{\mathbb Q}[x]\text{,} then p(x)p(x) factors as (x2−2)(x2−3).(x^2 - 2)(x^2 - 3)\text{.} However, both of these factors are irreducible in Q[x].{\mathbb Q}[x]\text{.} If we wish to find a zero of p(x),p(x)\text{,} we must go to a larger field. Certainly the field of real numbers will work, since

p(x)=(x−2)(x+2)(x−3)(x+3).\begin{equation*} p(x) = (x - \sqrt{2} ) (x + \sqrt{2} )( x - \sqrt{3})(x + \sqrt{3}). \end{equation*}

It is possible to find a smaller field in which p(x)p(x) has a zero, namely

Q(2)={a+b2:a,b∈Q}.\begin{equation*} {\mathbb Q }( \sqrt{2} ) = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \}. \end{equation*}

We wish to be able to compute and study such fields for arbitrary polynomials over a field F.F\text{.}