# Tag Info

## Hot answers tagged cyclotomic-fields

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### Why is the regulator of the degree 11 extension of $\mathbb{Q}$ so large?

A prime $\ell$ splits in $K_p$ for $p$ odd if and only if $$\ell^{p-1} \equiv 1 \mod p^2.$$ We can express $\lim_{s \to 1} (s-1) \zeta_{K_n(s)}$ as the product of $\frac{1}{ 1- |\mathfrak a|^{-1} }$ ...
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### Special units in the $11$th cyclotomic field

Yes. Indeed $$(1 + \zeta + \zeta^{10}) \, (\zeta + \zeta^4 + \zeta^7 + \zeta^{10}) = 1$$ with $\sum_i a_i = 3$ and $\sum_i b_i = 4$; now change each $a_i$ to $a_i+3$ and each $b_i$ to $b_i+3$. (...
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### Can a sum of roots of unity be an integer?

If $n$ is square-free this cannot happen (even just for the case $k=n$), and if $n$ is not square-free (and in the case $k=n$) one must have the sum being zero if it is an integer (as seen in Peter ...
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EDITED Despite the upvotes, the first version of this answer was mostly wrong, and I cannot delete it since it has been accepted. Let me see what can be salvaged. Fix $\zeta_n$, identify $(\mathbb{Z}... 7 votes Accepted ### Motivation for cyclotomic units I'm quite sure cyclotomic units were first introduced by Kummer in his 1847 paper, where he proved his very famous partial solution to Fermat's Last Theorem: Theorem (Kummer) If$p$is an odd prime ... • 17.1k 6 votes ### Special units in the$11$th cyclotomic field If I did this right there are a total of$1045 = 55 \cdot 19$solutions, obtained from the following$19$basic solutions by changing$a_i,b_i$to$a_{ri+s\bmod 11}$and$b_{ri+s \bmod 10}$for all$s\...
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I made hand calculations for $n\le4$. It turns out that even with $V_n=1_n$, the factor $U_n$ is not unique. In other words, the factorisation problem is underdetermined. This is due to the fact that ...