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\title{Notebook Problem 4.8 (Stewart)}
\author{Jason Wojciechowski}

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{\bf Theorem:} Let {\bf A} be the field of algebraic numbers and {\bf
  Q} be the field of rational numbers. Then $[{\bf A:Q}] = \infty$.

{\bf Proof:} We'll use Eisenstein's criterion to show that there exist
irreducible polynomials over {\bf Q} of arbitrarily large degree, thus
showing that $[{\bf A:Q}] = \infty$.

Take $f(x) = a_0 + a_1x + a_2 x^2 + \ldots + a_nx^n$ for any $n \in
{\bf N}$. Now choose a $q > 1$ such that $q \not| a_n$ (we know this
can always be found since we can just choose a $q$ greater than $a_n$)
and set $a_0,\ldots,a_{n-1}$ equal to $q$. Then
$q+qx+qx^2+\ldots+a_nx^n$ is irreducible by Eistenstein's criterion
since $q \not| a_n$, $q\; |\; a_i, 0 \leq i \leq n-1$, and $q^2 \not|
a_0$, since $q^2 \not| q$.

Thus, since we can raise the degree of $f(x)$ to arbitrarily large
degree, the extension $[{\bf A:Q}]$ can have arbitrarily large degree.

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