  The Brauer-Siegel theorem concerns the size of the product of the class
number and the regulator of a number field $K$. We derive bounds for this
product in case $K$ is a prime cyclotomic field, distinguishing between whether
there is a Siegel zero or not. In particular, we make a result of Tatuzawa
(1953) more explicit. Our theoretical advancements are complemented by
numerical illustrations that are consistent with our findings.
