Background in Basic Field Theory

Let $K$ be a field (in this post we mostly assume that $K\supset \mathbb{Q}$) and $n$ an integer $>1$ which is not divisible by the characteristic of $K$. Then the polynomial

is separable because its derivative is $n{X}^{n-1}\ne 0$. Hence in the algebraic closure $\bar{K}$, the polynomial has $n$ distinct roots, which forms a group $U$, and is cyclic. In fact, as an exercise, one can show that, for a field $k$, any subgroup $U$ of the multiplicative group $k^{\ast }$ is a cyclic group.

The generator ${\zeta }_n$ of $U$ is called the primitive $n$-th root of unity. Let $K=\mathbb{Q}$ and $L$ be the smallest extension that contains all elements of $U$, then we have $L=\mathbb{Q}({\zeta }_n)$. As a matter of fact, $L/K$ is a Galois extension (to be shown later), and the cyclotomic polynomial ${\mathrm{\Phi }}_n(X)$ is the irreducible polynomial of ${\zeta }_n$ over $\mathbb{Q}$. We first need to find the degree $[L:K]$.

Proposition 1. Notation being above, $L/K$ is Galois, the Galois group $\mathrm{Gal}(L/K)\cong (\mathbb{Z}/n\mathbb{Z}{)}^{\ast }$ (the group of units in $\mathbb{Z}/n\mathbb{Z}$) and $[L:K]=\phi (n)$.

Let’s first elaborate the fact that $|(\mathbb{Z}/n\mathbb{Z}{)}^{\ast }|=\phi (n)$. Let $[0],[1],\dots ,[n-1]$ be representatives of $\mathbb{Z}/n\mathbb{Z}$. An element $[x]$ in $\mathbb{Z}/n\mathbb{Z}$ is a unit if and only if there exists $[y]$ such that $[xy]=[1]$, which is to say, $xy\equiv 1\mathrm{mod}n$. Notice that $xy\equiv 1\mathrm{mod}n$ if and only if $xy+mn=1$ for some $y,n\in \mathbb{Z}$, if and only if $gcd(x,n)=1$. Therefore $|(\mathbb{Z}/n\mathbb{Z}{)}^{\ast }|=\phi (n)$ is proved.

The proof can be produced by two lemmas, the first of which is independent to the characteristic of the field.

Lemma 1. Let $k$ be a field and $n$ be not divisible by the characteristic $p$. Let $\zeta ={\zeta }_n$ be a primitive $n$-th root of unity in $\bar{k}$, then $(\mathbb{Z}/n\mathbb{Z}{)}^{\ast }\supset \mathrm{Gal}(k(\zeta )/k)$ and therefore $[k(\zeta ):k]\le \phi (n)$. Besides, $k(\zeta )/k$ is a normal abelian extension.

Proof. Let $\sigma$ be an embedding of $k(\zeta )$ in $\bar{k}$ over $k$, then

so that $\sigma \zeta$ is also an $n$-th root of unity also. Hence $\sigma \zeta ={\zeta }^i$ for some $i=i(\sigma )$, uniquely determined modulo $n$. It follows that $\sigma$ maps $k(\zeta )$ into itself. This is to say, $k(\zeta )$ is normal over $k$. Let $\tau$ be another automorphism of $k(\zeta )$ over $k$ then

It follows that $i(\sigma )$ and $i(\tau )$ are prime to $n$ (otherwise, $\sigma \zeta$ would have a period smaller than $n$, implying that the period of $\zeta$ is smaller than $n$, which is absurd). Therefore for each $\sigma \in \mathrm{Gal}(k(\zeta )/k)$, $i(\sigma )$ can be embedded into $(\mathbb{Z}/n\mathbb{Z}{)}^{\ast }$, thus proving our theorem. $◻$

It is easy to find an example with strict inclusion. One only needs to look at $k=\mathbb{R}$ or $\mathbb{C}$.

Lemma 2. Let $\zeta ={\zeta }_n$ be a primitive $n$-th root of polynomial over $\mathbb{Q}$, then for any $p\nmid n$, ${\zeta }^p$ is also a primitive $n$-th root of unity.

Proof. Let $f(X)$ be the irreducible polynomial of $\zeta$ over $\mathbb{Q}$, then $f(X)|(X^n-1)$ by definition. As a result we can write $X^n-1=f(X)h(X)$ where $h(X)$ has leading coefficient $1$. By Gauss’s lemma, both $f$ and $h$ have integral coefficients.

Suppose ${\zeta }^p$ is not a root of $f$. Since $({\zeta }^p{)}^n-1=({\zeta }^n{)}^p-1=0$, it follows that ${\zeta }^p$ is a root of $h$, and $\zeta$ is a root of $h(X^p)$. As a result, $f(X)$ divides $h(X^p)$ and we write

Again by Gauss’s lemma, $g(X)$ has integral coefficients.

Next we reduce these equations in ${\mathbf{F}}_p=\mathbb{Z}/p\mathbb{Z}$. We firstly have

By Fermat’s little theorem $a^p=a$ for all $a\in {\mathbf{F}}_q$, we also have

Therefore

which implies that $\bar{f}(X)|\bar{h}(X{)}^p$. Hence $\bar{f}$ and $\bar{h}$ must have a common factor. As a result, $X^n-\bar{1}=\bar{f}(X)\bar{h}(X)$ has multiple roots in ${\mathbf{F}}_p$, which is impossible because of our choice of $p$. $◻$

Now we are ready for Proposition 1.

Proof of Proposition 1. Since $\mathbb{Q}$ is a perfect field, $\mathbb{Q}(\zeta )/\mathbb{Q}$ is automatically separable. This extension is Galois because of lemma 1. By lemma 1, it suffices to show that $[\mathbb{Q}(\zeta ):\mathbb{Q}]\ge \phi (n)$.

Recall in elementary group theory, if $G$ is a finite cyclic group of order $m$ and $x$ is a generator of $G$, then the set of generators consists elements of the form $x^{\nu }$ where $\nu \nmid m$. In this occasion, if $\zeta$ generates $U$, then ${\zeta }^p$ also generates $U$ because $p\nmid n$. It follows that every primitive $n$-th root of unity can be obtained by raising $\zeta$ to a succession of prime numbers that do not divide $n$ (as a result we obtain exactly $\phi (n)$ such primitive roots). By lemma 2, all these numbers are roots of $f$ in the proof of lemma 2. Therefore $\mathrm{deg}f=[L:K]\ge \phi (n)$. Hence the proposition is proved. $◻$

We will show that $f$ in the proof lemma 2 is actually the cyclotomic polynomial ${\mathrm{\Phi }}_n(x)$ you are looking for. The following procedure works for all fields where the characteristic does not divide $n$, but we assume characteristic to be $0$ for simplicity.

We have

where the product is taken over all $n$-th roots of unity. Collecting all roots with the same period $d$ (i.e., those $\zeta$ such that ${\zeta }^d=1$), we put

Then

It follows that ${\mathrm{\Phi }}_1(X)=X-1$ and

This presentation makes our computation much easier. But to understand ${\mathrm{\Phi }}_n$, we still should keep in mind that the $n$-th cyclotomic polynomial is defined to be

whose roots are all primitive $n$-th roots of unity. As stated in the proof of proposition 1, there are $\phi (n)$ primitive $n$-th roots of unity, and therefore $\mathrm{deg}{\mathrm{\Phi }}_n(X)=\phi (n)$. Besides, $f|{\mathrm{\Phi }}_n$. Since both have the same degree, these two polynomials equal. It also follows that $\sum _{d|n}\phi (n)=n$.

Proposition 2. The cyclotomic polynomial is irreducible and is the irreducible polynomial of $\zeta$ over $\mathbb{Q}$, where $\zeta$ is a primitive $n$-th root of unity.

We end this section by a problem in number fields, making use of what we have studied above.

Problem 0. A number field $F$ only contains finitely many roots of unity.

Solution. Let $\zeta \in F$ be a root of unity with period $n$. Then ${\mathrm{\Phi }}_n(\zeta )=0$ and therefore $[\mathbb{Q}(\zeta ):\mathbb{Q}]$ has degree $\phi (n)$. Since $\mathbb{Q}(\zeta )$ is also a subfield of $F$, we also have $\phi (n)\le [F:\mathbb{Q}]$. Since $\{n:\phi (n)\le [F:\mathbb{Q}]\}$ is certainly a finite set, the number of roots of unity lie in $F$ is finite. $◻$

Technical Computations

We will do some dirty computation in this section.

Problem 1. If $p$ is prime, then ${\mathrm{\Phi }}_p(X)=X^{p-1}+X^{p-2}+\cdots +1$, and for an integer $\nu \ge 1$, ${\mathrm{\Phi }}_{p^{\nu }}(X)={\mathrm{\Phi }}_p(X^{p^{\nu -1}})$.

Solution. The only integer $d$ that divides $p$ is $1$ and we can only have

For the second statement, we use induction on $\nu$. When $\nu =1$ we have nothing to prove. Suppose now

is proved, then $X^{p^{\nu }}-1=\prod _{r=0}^{\nu }{\mathrm{\Phi }}_{p^r}(X)$ and therefore

Problem 2. Let $p$ be a prime number. If $p\nmid n$, then

$${\mathrm{\Phi }}_{pn}(X)=\frac{{\mathrm{\Phi }}_n(X^p)}{{\mathrm{\Phi }}_n(X)}.$$

Solution. Assume $p\nmid n$ first. It holds clearly for $n=1$. Suppose now the statement holds for all integers $<n$ that are prime to $p$. We see

Problem 3. If $n$ is an odd number $>1$, then ${\mathrm{\Phi }}_{2n}(X)={\mathrm{\Phi }}_n(-X)$.

Solution. By problem 2, ${\mathrm{\Phi }}_{2n}(X)={\mathrm{\Phi }}_n(X^2)/{\mathrm{\Phi }}_n(X)$. To show the identity it suffices to show that

For $n=3$ we see

Now suppose it holds for all odd numbers $3\le d<n$, then

The following problem would not be very easy without the Möbius inversion formula so we will use it anyway. Problems above can also be deduced from this formula. Let $f:{\mathbb{Z}}_{\ge 0}\to {\mathbb{Z}}_{\ge 0}$ be a function and $F(n)=\prod _{d|n}f(d)$, then the Möbius inversion formula states that

with

Putting $f(d)={\mathrm{\Phi }}_d(X)$, we see

Now we proceed.

Problem 4. If $p|n$, then ${\mathrm{\Phi }}_{pn}(X)={\mathrm{\Phi }}_n(X^p)$.

Solution. By the Möbius inversion formula, we see

because all $d$ that divides $np$ but not $n$ must be divisible by $p^2$. Problem 2 can also follow from here.

Problem 5. Let $n=p_1^{r_1}\dots p_s^{r_s}$, then

$${\mathrm{\Phi }}_n(X)={\mathrm{\Phi }}_{p_1\dots p_s}(X^{p_1^{r_1-1}\dots p_s^{r_s-1}}).$$

Solution. This problem can be solved by induction on the number of primes. For $s=1$ it is problem 1. Suppose it has been proved for $s-1$ primes, then for

and a prime $p_s$, we have

On the other hand,

if we put $Y=X^{p_1^{r_1-1}\dots p_{s-1}^{r_{s-1}-1}}$. When it comes to higher degree of $p_s$, it’s merely problem 2. Therefore we have shown what we want.

Computing the Norm

Let $\zeta$ be a primitive $n$-th root of unity, put $K=\mathbb{Q}(\zeta )$ and $G$ the Galois group.. We will compute the norm of $1-\zeta$ with respect to the extension $K/\mathbb{Q}$. Since this extension is separable, we have

Since $G$ acts on the set of primitive roots transitively, $\{\sigma \zeta {\}}_{\sigma \in G}$ is exactly the set of primitive roots of unity, which are roots of ${\mathrm{\Phi }}_n(X)$. It follows that

If $n=p^r$, then $N_{\mathbb{Q}}^K(1-\zeta )={\mathrm{\Phi }}_p(1^{p^{r-1}})={\mathrm{\Phi }}_p(1)=p$. On the other hand, if

then