Cyclotomic fields are fields where a primitive root of unity is added to the rational numbers. These fields are common in algebraic number theory.
Questions tagged [cyclotomic-fields]
410 questions
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Interesting product in a cyclotomic field
Let $n$ be an integer greater $2$. What is the value of the following product for $a=3, \dots , n-1$:
$$\prod_{k = 1}^{n-1} \left( 1 - \sum_{j = 1}^{a-1} \zeta^{jk} \right)$$
where $\zeta$ is some complex $n$th root of unity. I can prove…
Marcin
- 11
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Question about section 9.1 of Washington's "cyclotomic fields"
I am trying to understand the Basic Argument from Chapter 9.1 of Washington's "cyclotomic fields", and I can understand all but one part. Under Assumption 1: $p \nmid h^{+}(\mathbb{Q}(\zeta_p))$ it says:
"Note that $\overline{B_0} = B_0$ and…
lewy
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The Ideal Class Group of $R = \mathbb{Z}[e^{2i\pi/p}]$
I wish to show that ideal class group of $R$ is finite. To accomplish this I am given several parts to prove. I have proven some of them but not all. These are the things I am stuck on.
Suppose A is an ideal of $R.$ How do I show that there exists…
Debbie
- 840
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Generator for cyclotomic units
In the proof of Theorem 7.2 in the Appendix by Rubin on the Iwasawa Main Conjecture in Lang's Cyclotomic Fields, it is claimed that
For an even nontrivial character $\chi$ of $\Delta$, $\xi^{e(\chi)}$ generates $e(\chi)V_n$ where
for a character…
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class number for special cyclotomic fields
Are the class number of the cyclotomic fields $Q(\mu_n)$ known, where $n=2^k$ ? Especially for $k=12$ ?
Such fields are often used in Post Quantum cryptography, e.g. in Ring Learning With Errors systems.
user274886
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Prove that tan(Pi/n)/tan(kPi/n) is an algebraic integer when gcd(k,n) = 1
This is equivalent to showing that $\left[ \frac{\zeta -1}{\zeta +1} \right]\left[ \frac{{{\zeta }^{k}}+1}{{{\zeta }^{k}}-1} \right]$ is an integer in the cyclotomic field $\text{Q(}\zeta )$ where $\zeta ={{e}^{2\pi i/n}}$ and gcd(k,n) = 1.
sunshineghh
- 121
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Basis of the n-th cyclotomic field over Q
How can one find a basis of the n-th cyclotomic field as a vector space over Q in a standard way ?
Thank you !
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Characterization of the complement of the maximal real subfield of a cyclotomic field $\mathbb{Q}(\zeta )$.
If $\zeta$ is a primitive nth root of unity, it appears that the cyclotomic field $\mathbb{Q}(\zeta )$can be partitioned into ${\mathbb{Q}}(\zeta +{{\zeta }^{-1}})$ and ${\mathbb{Q}}(\zeta -{{\zeta }^{-1}})$ where ${{\mathbb{Q}}^{+}}$=…
sunshineghh
- 121