4 – Cyclotomic fields

(This is part of my series on algebraic number theory.)

In this post we present a special class of number fields called cyclotomic fields which are of particular interest. Very thick volumes have been written on cyclotomic fields alone, for instance by Lang or, on a more introductory level, by Washington.

4.1 Cyclotomic fields

Recall that a number field is a field of the form $\mathbf{Q}(\alpha)$, for an element $\alpha$ algebraic over $\mathbf{Q}$. That is, $\alpha$ is a root of a polynomial with coefficients in $\mathbf{Q}$, and so we see that $\alpha$ must be in $\mathbf{C}$ (since a polynomial with rational coefficients can also be viewed as a polynomial in complex coefficients, and a polynomial with complex coefficients has all its roots in $\mathbf{C}$).

Of particular interest are the fields $\mathbf{Q}(\alpha)$ where $\alpha$ is a primitive $n$th root of unity in $\mathbf{C}$. $\alpha$ is a $n$th root of unity if $\alpha^n = 1$, and is a primitive $n$th root of unity if in addition $\alpha^k \ne 1$ for any $0 < k < n$ (that is, if $n$ is the smallest positive integer such that $\alpha^n = 1$). Examples:

  • The square roots of unity in $\mathbf{C}$ are $1$ and $-1$. Of those, only $-1$ is a primitive square root of unity since for $1$ we have $1^1 = 1$.
  • The $4$th roots of unity in $\mathbf{C}$ are $\pm 1$ and $\pm i$. Of those, only $\pm i$ are primitive $4$th roots of unity.

In general, a primitive $n$th root of unity is always given by $e^{2i\pi/n}$. A primtive $n$th root of unity is also sometimes noted $\zeta_n$, when we are not concerned with its precise value but only with the fact that it is a primitive $n$th root of unity.

If $\alpha$ is a primitive $n$th root of unity, the field $\mathbf{Q}(\alpha)$ is called the $n$th cyclotomic field. We are justified in saying the $n$th cyclotomic field because the precise primitive $n$th root of unity we add to $\mathbf{Q}$ is unimportant: we obtain the same field no matter which one we add.

Examples:

  • Since $-1$ is a primitive square root of unity, the $2$nd cyclotomic field is $\mathbf{Q}(-1)$. Since $-1$ is in $\mathbf{Q}$ we have $\mathbf{Q}(-1) = \mathbf{Q}$ and we see that the $2$nd cyclotomic field is just $\mathbf{Q}$ itself.
  • The $4$th cyclotomic field is $\mathbf{Q}(i) = \mathbf{Q}(-i)$. Indeed, by the property of additive inverse, since $\mathbf{Q}(i)$ is a field which contains $i$ it must also contain $-i$, and conversely since $\mathbf{Q}(-i)$ is a field which contains $-i$ it must also contain $i$. This means that in the end, both $\mathbf{Q}(i)$ and $\mathbf{Q}(-i)$ can be defined as the smallest field containing $\mathbf{Q}$, $i$, and $-i$.

4.2 Cyclotomic polynomials

Let $\mathbf{Q}(\alpha)$ be the $n$th cyclotomic field. In order to have a precise idea of what it is, we must obtain the minimal polynomial of $\alpha$ over $\mathbf{Q}$. This polynomial is called the $n$th cyclotomic polynomial. Computing it in general is not an easy task (see Wikipedia for some examples of cyclotomic polynomials for various values of $n$), but fortunately the case in which we are most interested is much simpler.

We are interested in the $p$th cyclotomic field, where $p$ is an odd prime (the case $p=2$ is trivial). Recall that for $\alpha$ a primitive $p$th root of unity we have $\alpha^p = 1$, and so $\alpha$ is a root of the polynomial $x^p-1$. Since $\alpha \ne 1$, it follows that it is a root of
\[ \frac{x^p-1}{x-1} = x^{p-1} + x^{p-2} + \dots + x + 1, \]
and it can be shown that if $p$ is prime, this polynomial is irreducible over $\mathbf{Q}$. Thus it is precisely the $p$th cyclotomic polynomial, generally noted $\Phi_p$ (note that $\Phi_p$ is thus of degree $p-1$).

Thus $\alpha$ has degree $p-1$ over $\mathbf{Q}$, and the elements of the $p$th cyclotomic field $\mathbf{Q}(\alpha)$ are the elements of the form
\[ a_0 + a_1\alpha + \dots + a_{p-2}\alpha^{p-2}, \]
where all the coefficients $a_i$ are in $\mathbf{Q}$.

Next: Computing in number fields

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