Global Theory

  1. Define the ring of integers of a number field. Give a characterization of the ring of integers of quadratic and cyclotomic extensions of $ {\mathbb{Q}}$. Prove the result for quadratic extensions.

  2. What is an integral basis? What is the norm of an ideal and why is it finite?

  3. Define the terms ramification index and inertia degree. Which primes ramify in quadratic and cyclotomic extensions of $ {\mathbb{Q}}$?

  4. Let $ L/K$ be an extension of number fields, $ {\mathfrak{p}}\subset \O _K$ a prime. Factor $ {\mathfrak{p}}\O _L$ as a product of primes $ \P _1^{e_1} \cdots
\P _r^{e_r}$. What is the relation of the $ e_i$,$ r$ and $ [L:K]$. If $ L/K$ is Galois, how does this affect things and why? Suppose $ \O _L
= \O _K[\alpha]$. How would you determine the $ \P _i$?

  5. Let $ p \equiv 1\pmod 4$ be a prime, and let $ K = {\mathbb{Q}}(\sqrt p)$. Determine all primes of $ {\mathbb{Q}}$ which ramify in $ K$. Determine congruence conditions which describe all primes $ q$ which split (completely) in $ K$, and congruence conditions for those primes which are inert.

  6. Let $ p \equiv 1\pmod 4$ be a prime, $ K = {\mathbb{Q}}(\sqrt p)$, and $ L
= {\mathbb{Q}}(\zeta_p)$ be the $ p$th cyclotomic extension. Show that $ K
\subset L$. Let $ q$ be an odd prime which satisfies $ q^r \equiv -1
\pmod p$ for some odd positive integer $ r$. Determine the factorization (type) of $ q\O _K$ and $ q\O _L$. How do the primes which occur in the factorization of $ q\O _K$ factor in $ \O _L$? Work this out explicitly with $ p=13$ and $ q = 17$.

  7. Let $ L/K$ be a Galois extension of number fields, $ {\mathfrak{p}}\subset \O _K$ a prime. Let $ \P\subset \O _L$ be a prime lying above $ {\mathfrak{p}}$. Describe the decomposition and inertia groups associated to $ {\mathfrak{p}}$ and $ \P$. Determine the order of the decomposition group. If $ \P '$ is another prime of $ L$ lying above $ {\mathfrak{p}}$, how are the decomposition groups $ D(\P\mid {\mathfrak{p}})$ and $ D(\P '\mid {\mathfrak{p}})$ related? Suppose that $ {\mathfrak{p}}$ is unramified in $ L$. Describe how to construct the Frobenius map $ \displaystyle \left[\frac{L/K}{\P }\right]$. Show that for abelian extensions, the Frobenius map depends only on $ {\mathfrak{p}}$ and not on $ \P$.

  8. Let $ K = {\mathbb{Q}}(\zeta_{13})$, $ {\mathfrak{p}}= 17{\mathbb{Z}}$, $ \P$ any prime of $ K$ lying above $ {\mathfrak{p}}$ and $ D$ and $ T$ the associated decomposition and inertia groups. Let $ K_D$ and $ K_T$ be the associated fixed fields. Determine them.

  9. Show that $ {\mathbb{Q}}(\sqrt{-5})$ has class number 2 using the Minkowski bound.

root 2007-06-06