Theorem 1.7.1 If $f(x)=a_0+a_1x+\cdots+a_nx^n\in \mathbb Z[x]\subset \mathbb Q[x]$, then every rational root of $f$ has the form $b/c$, where $b\mid a_0$ and $c\mid a_n$. In particular, if $f$ is monic, then every rational root of $f$ is an integer.

Theorem 1.7.2 Let $f(x)=a_0+a_1x+\cdots+a_{n-1}x^{n-1}+x^n\in \mathbb Z[x]$ be monic, and let $p$ be a prime. If $\overline{f}(x)=[a_0]+[a_1]x+\cdots+[a_{n-1}]x^{n-1}+x^n$ is irreducible in $\mathbb F_p[x]$, then $f$ is irreducible in $\mathbb Q[x]$.

Definition 1.7.1 If $n\ge 1$ is a positive integer, then an $n$th root of unity in a field $k$ is an element $\zeta\in k$ with $\zeta^k=1$. If $\zeta$ is an $n$th root of unity and $n$ is the smallest positive integer for which $\zeta^n=1$, we say that $\zeta$ is a primitive $n$th root of unity.

Definition 1.7.2 If $d$ is a positive integer, then the $d$th cyclotomic polynomial is defined by
$$\Phi_d(x)=\prod(x-\zeta),$$
where $\zeta$ ranges over all the primitive $d$th roots of unity.

Proposition 1.7.1 Let $n$ be a positive integer and regard $x^n-1\in \mathbb Z[x]$. Then

1. $$x^n-1=\prod{d\mid n}\Phi{d}(x),$$

where $d$ ranges over all the positive divisors $d$ of $n$.

2. $\Phi_n (x)$ is a monic polynomial in $\mathbb Z[x]$ and $\deg(\Phi_n)=\phi(n)$, the Euler $\phi$-function.

3. For every integer $MARKDOWN_HASH1d0fe3ad28ed27546808efeec5c626a9MARKDOWNHASH$, we have
$$n=\sum {d\mid n}\phi(d).$$

Proof.

1. For each divisor $d$ of $n$, collect all terms in the equation $x^n-1=\prod(x-\zeta)$ with $\zeta$ a primitive $d$th root of unity.

2. Let's prove that $\Phi_n(x)\in \mathbb Z[x]$ by induction on $n\ge1$. The base step is true, for $\Phi_1(x)=x-1\in \mathbb Z[x]$.

For the inductive step, let $f(x)=\prod_{d\mid n,d<n}\Phi_d(x)$, so that
$$x^n-1=f(x)\Phi_n(x).$$
By induction, each $\Phi_d(x)$ is a monic polynomial in $\mathbb Z[x]$, and so $f$ is a monic polynomial in $\mathbb Z[x]$.

Since $f$ and $x^n-1$ are monic, $\Phi_n$ is a monic polynomial in $\mathbb Z[x]$.

3. Immediate from parts (1) and (2):
$$n=\deg(x^n-1)=\deg(\prod_d \Phi_d)=\sum_d \deg(\Phi_d)=\sum_d \phi(d)\ \square$$

Corollary 1.7.1 If $q$ is a positive integer and $d$ is a divisor of an integer $n$ with $d<n$, then $\Phi_n(q)$ is a divisor of both $q^n-1$ and $(q^n-1)/(q^d-1)$.

Theorem 1.7.3 (Eisenstein Criterion) Let $f(x)=a_0+a_1x+\cdots+a_nx^n\in \mathbb Z[x]$. If there is a prime $p$ dividing $a_i$ for all $i<n$ but with $p\nmid a_n$ and $p\nmid a_0^2$, then $f$ is irreducible in $\mathbb Q[x]$.

Theorem 1.7.4 (Gauss) For every prime $p$, the $p$th cyclotomic polynomial $\Phi_p(x)$ is irreducible in $\mathbb Q[x]$.

Proof.

Since $\Phi_p(x)=(x^p-1)/(x-1)$, we have
$$\Phi(x+1)=((x+1)^p-1)/x=x^{p-1}+C_p^1x^{p-2}+C_p^2x^{p-3}+\cdots+p.$$
Since $p$ is prime, we have $p\mid C_p^i$ for all $i\ (0<i<p)$; hence, Eisenstein Criterion applies, and $\Phi(x+1)$ is irreducible in $\mathbb Q[x]$. $\square$

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