If we consider the polynomial \(p(x)=x^7 + 12x^3 + 2x +6 \in
\Z[x]\text{,}\) we know it is irreducible by Eisenstein’s criterion (\(p=2\)), so in particular, \(p\) has no roots in \(\Q.\) Where are its roots? Is there a smallest field containing all the roots of \(p\text{?}\)
Perhaps this seems an uninteresting question because somewhere in your distant past, you learned that all the roots of \(p\) must be contained in \(\C\) since \(\C\) is algebraically closed. Well, for starters, the complex numbers is an enormous field (certainly uncountable), and are you really all that comfortable with how to construct it?
In fact all the roots of all the polynomials in \(\Q[x]\) are contained in a subfield which is countable, and the smallest field containing the roots of any given polynomial is only a finite-dimensional vector space over the rationals.
Now add to that the following question. Consider the polynomial \(x^2 -3\in (\Z/5\Z)[x] \text{.}\) It is irreducible since it has degree two and no roots in the field \(\Z/5\Z\text{.}\) Is there a field containing its roots? Certainly the real and complex numbers are of no help here since this field has characteristic 5. So the ability to construct such a field is really significant.
If \(K\) is a field and \(p\in K[x]\) is irreducible, we know that \((p)\) is a maximal ideal, and hence \(K[x]/(p)\) is a field. We shall show that this is a field containing an isomorphic copy of \(K\) and a root of the polynomial \(p.\)
Let’s consider a motivating example. We want to construct a field \(K\) containing \(\Q\) which also contains a root of the polynomial \(p(x)=x^7 -10.\) Eisenstein’s criterion (with \(p=2,5\)) assures us this polynomial is irreducible in \(\Q[x],\) so in particular has no roots in \(\Q.\) We claim that \(\Q[x]/(x^7-10)\) is such a field \(K.\)
Now of course since this is motivation, we shall acknowledge the existence of the real and complex numbers, and let \(\root7\of{10}\) denote any such root in \(\C.\) The polynomial has one real and six complex roots. Consider the evaluation map
given by \(\varphi(f(x)) = f(\root7\of {10}),\) and the figure below.
Figure3.6.1.A field containing a root of \(x^7-10\)
The map \(\varphi\) is a surjective homomorphism with image \(\Q[\root7\of{10}]\) which is certainly an integral domain as a subring of the field \(\C.\) Moreover, it is clear that the ideal \((x^7-10)\subseteq \ker \varphi\text{,}\) but also we know \((x^7-10)\) is a maximal ideal since \(x^7 -10\) is irreducible in the PID \(\Q[x]\) (see Proposition 3.4.7), and since \(\varphi\) is not the zero homomorphism, it follows that \(\ker \varphi = (x^7-10).\) Thus we conclude
Under the isomorphism \(\varphi_*\text{,}\) the element \(\alpha = x+(x^7-10) \in \Q[x]/(x^7-10)\) corresponds to \(\root7\of{10} \in \Q[\root7\of{10}]\text{.}\) But what does that really mean?
First, let’s understand how \(\Q\) is a subfield of \(\Q[x]/(x^7-10)\text{.}\) Consider the map
which takes \(a\in
\Q\) to the coset \(a+(x^7-10).\) In particular
\begin{equation*}
10 \mapsto \text{ the coset }10+(x^7-10)\text{.}
\end{equation*}
So what is \(\alpha^7 - 10 \in \Q[x]/(x^7-10)\text{,}\) meaning what is \(\alpha^7 - [10+(x^7-10)]\text{?}\) For compact notation, let \(I = (x^7-10).\)
Which says the element \(\alpha = x+I \in K=\Q[x]/I\) is the seventh root of 10.
Remark3.6.2.
The example above is entirely representative of the general situation. If \(F\) is any field, and \(p \in
F[x]\) is irreducible, then \(K := F[x]/(p)\) is a field containing an isomorphic copy of \(F\) and a root \(\alpha = x+(p(x))\) of the polynomial \(p\text{,}\) and to ease the notation, we generally denote the field \(K\) not as the quotient, but as \(K=F(\alpha)\) which denotes the smallest field containing (an isomorphic copy of) \(F\) and the root \(\alpha.\)
When first seeing this, one is probably more in their comfort zone seeing things like \(\Q(\sqrt 2)\) or \(\Q(\root7\of {10})\text{,}\) but when you think harder, if the polynomial is something like \(p(x)=x^7 + 12x^3 + 2x
+6 \) which is irreducible in \(\Q[x]\text{,}\) what better notation would there be than simply to write \(\Q(\alpha),\) where \(\alpha\) is some root of \(p.\) The point of the construction is that we know that we can construct such a field and an appropriate root.
Remark3.6.3.
One might have noticed that in the example of the construction, the notation \(\Q[\root7\of{10}]\) (square brackets) was used instead of the later notation such as \(\Q(\sqrt 2)\) (parentheses).
There is a distinction between the notations \(F[\alpha]\) (the smallest ring containing \(F\) and \(\alpha\)) in contrast to \(F(\alpha)\) (the smallest field containing \(F\) and \(\alpha\)) which involves a discussion of whether \(\alpha\) is algebraic or transcendental over \(F\text{,}\) but in the cases above they are equal.
There is however a useful observation to be made. First we observed that \(F[\alpha] \cong F[x]/(p)\) was a field since \((p)\) is maximal, so that \(F[\alpha]\) actually equals \(F(\alpha)\text{,}\) the smallest field containing \(F\) and \(\alpha.\)
The beauty of knowing that \(F[\alpha]\) is a field is that every element of the field can be written as \(a_0 + a_1 \alpha + \cdots + a_n\alpha^n\) for some \(n\) and \(a_i \in F.\) In particular, an element like \(1/\alpha\) has such a representation!
We conclude this section on applications by giving a characterization of \(F[\alpha]\) when \(\alpha\) is a root of some irreducible polynomial \(p\) with coefficients in \(F.\)
Theorem3.6.4.
Let \(F\) be a field and \(p\in F[x]\) an irreducible polynomial of degree \(d\text{.}\) Then \(F[x]/(p)\) is a field which as a vector space over \(F\) has dimension \(d\) with basis \(\{\overline 1,
\overline x, \dots, \overline x^{d-1}\}.\) Here, we use the notation \(\overline x^j = x^j + (p).\)
That means that when we write \(F[\alpha]\cong
F[x]/(p),\) every element of the field \(F[\alpha]\) has a representation as \(a_0 + a_1 \alpha + \cdots + a_{d-1}\alpha^{d-1}\) for uniquely determined \(a_i \in F.\)
Proof.
By the division algorithm, every element \(f(x) \in
F[x]\) can be written as
where either \(r = 0\) or \(\deg r \lt \deg p=d.\) This means that \(f(x) + (p) = r(x) + (p).\) Thus it is clear that \(\{\overline 1,
\overline x, \dots, \overline x^{d-1}\}\) spans \(F[x]/(p)\) as a vector space over \(F.\)
To show independence, suppose that
\begin{equation*}
a_0 + a_1 \overline x + \cdots + a_{d-1}\overline x^{d-1}
= \overline 0 \text{ in } F[x]/(p).
\end{equation*}
