Constructing a field in which polynomial has root

• The natural map $\iota\colon F\longrightarrow F[x]/(g)$ defined by $\iota(a)=a+(g)$ is injective. So, $\iota(F)$ is isomorphic to $F$ and therefore, although $F$ is not actually a subset of $F[x]/(g)$, it is natural te see $F[x]/(g)$ as an extension of $F$.
• Because $g(\alpha)=g\bigl(x+(g)\bigr)=g(x)+(g)=(g)$, and $(g)$ is the $0$ element of $F[x]/(g)$.

Consider the sequence of canonical morphisms $F\to F[X] \to F[X]/(g) = F_1$. These are ring morphisms so the composite is as well : but a ring morphism between fields is precisely a field morphism; those are known to be injective: hence we actually have an embedding $F\to F_1$ which allows us to see $F_1$ as an extension of $F$.Call this embedding $\iota$, and the embedding $F\to F[X]$ is $i$: $\iota = \pi\circ i$

Now if $g= \displaystyle\sum_{k=0}^d a_kX^k$, then $\iota(g)(\alpha)= \displaystyle\sum_{k=0}^d\iota(a_k)\alpha^k = \displaystyle\sum_{k=0}^d\pi(i(a_k))\pi(X)^k = \pi(\displaystyle\sum_{k=0}^di(a_k)X^k)= \pi(g) = 0$. Hence $\alpha$ is a root of $\iota(g)$.

But if we identify $F$ and $\iota(F)$ through $\iota$ this precisely means that $\alpha$ is a root of $g$.

There's no magic happening here: we want to find a root of $g$ so we add an element to $F$ (we get $F[X]$) and declare :"this new element is a root of $g$" (we kill $(g)$ by modding it out)

There is a low budget way to do this. You may be familiar with taking the Gaussian integers as 2 by 2 matrices; this is the same procedure, but allowing rational coefficients. Now that I think of it, the same procedure with real number coefficients gives the complex numbers.

Suppose we have an irreducible polynomial over the rationals such as $$ g(x) = x^3 + x^2 - 2 x - 1. $$ We construct the "companion matrix" $$ A = \left( \begin{array}{rrr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 2 & -1 \end{array} \right) $$ The matrix $A$ will stand as the representative of your root $\alpha.$ The elements of the field are, given rational coefficients $u,v,w,$ just $$ uI + v A + w A^2 = \left( \begin{array}{rrr} u & v & w \\ w & u + 2 w & v - w \\ v - w & 2v - w & u - v + 3w \end{array} \right) $$

The rational numbers are into this matrix ring as $rI.$ The multiplicative identity is $I.$ Addition is obviously closed. Multiplication uses the identities $$ A^3 = I + 2 A - A^2 $$ $$ A^4 = -I - A + 3 A^2 $$ There are no zero divisors since $x^3 + x^2 - 2 x - 1$ was irreducible. Multiplication is commutative because everything is a power of $A.$

The ring is a field since can find the reciprocal of anything nonzero, for instance $$ A^{-1} = A^2 + A - 2 I. $$ Finally $\alpha$ is a root because of Cayley-Hamilton.