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prime ideal decomposition in cyclotomic extensions of 
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(Theorem)
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Let $q\in\Ints$ be a prime greater than $2$ , let $\zeta_q=e^{2\pi i/q}$ and write $L=\Rats(\zeta_q)$ for the cyclotomic extension. The ring of integers of $L$ is $\mathcal{O}_L=\Ints[\zeta_q]$ . The discriminant of $L/\Rats$ is: $$D_{L/\Rats}=\pm q^{q-2}$$ and it is $+$ exactly when $q-1\equiv 0,1\ \operatorname{mod}\ 4$
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Proposition 1 $\sqrt{\pm q}\in \Rats(\zeta_q)$ , with $+$ exactly when $q-1\equiv 0,1\ \operatorname{mod}\ 4$ .
Proof. It can be proved that: $$D_{L/\Rats}=\pm q^{q-2}=\prod_{1\leq i < j \leq q-1}(\zeta_q^i-\zeta_q^j)^2$$ Taking square roots we obtain $$q^{\frac{q-3}{2}}\sqrt{\pm q}=\prod_{1\leq i < j \leq q-1}(\zeta_q^i-\zeta_q^j)\in \Rats(\zeta_q)$$ Hence the result holds (and the sign depends on whether $q-1\equiv 0,1\ \operatorname{mod}\ 4$ ). 
Let $K=\Rats(\sqrt{\pm q})$ with the corresponding sign. Thus, by the proposition we have a tower of fields:
![$ \xymatrix{ & L=\mathbb{Q}(\zeta_q) \ar@{-}[d]\ & K \ar@{-}[d]\ & \mathbb{Q}}$](http://images.planetmath.org:8080/cache/objects/4644/js/img2.png "$ \xymatrix{ & L=\mathbb{Q}(\zeta_q) \ar@{-}[d]\ & K \ar@{-}[d]\ & \mathbb{Q}}$")
For a prime ideal $p\Ints$ the decomposition in the quadratic extension $K/\Rats$ is well-known (see this entry). The next theorem characterizes the decomposition in the extension $L/\Rats$ :
Theorem 1 Let $p\in\Ints$ be a prime.
- If $p=q$ , $q\mathcal{O}_L=\left(1-\zeta_q\right)^{q-1}$ . In other words, the prime $q$ is totally ramified in $L$ .
- If $p\neq q$ then $p\Ints$ splits into $(q-1)/f$ distinct primes in $\mathcal{O}_L$ , where $f$ is the order of $p\ \operatorname{mod}\ q$ (i.e. $p^f\equiv 1\ \operatorname{mod}\ q$ , and for all $1< n<f, p^n\neq 1\ \operatorname{mod}\ q$ ).
- 1
- Daniel A.Marcus, Number Fields. Springer, New York.
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"prime ideal decomposition in cyclotomic extensions of  " is owned by alozano.
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Cross-references: order, totally ramified, extension, theorem, quadratic extension, decomposition, prime ideal, fields, proposition, square roots, discriminant, ring of integers, cyclotomic extension, prime
There is 1 reference to this entry.
This is version 2 of prime ideal decomposition in cyclotomic extensions of  , born on 2003-08-22, modified 2005-12-28.
Object id is 4644, canonical name is PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ.
Accessed 2541 times total.
Classification:
| AMS MSC: | 11R18 (Number theory :: Algebraic number theory: global fields :: Cyclotomic extensions) |
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Pending Errata and Addenda
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