Number Theory

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Prove that the class group of a number field is finite. (Ribet)
What is your favorite proof of quadratic reciprocity? (Ribet)
Prove that there exists a field of order $p n$ for every prime $p$ and positive integer $n$ . (Peres (Stat))
Prove that ${\mathbb Q}[\sqrt{3}]$ has no unramified extensions. (Ribet)
Describe the ring of integers in ${\mathbb Q}(\zeta_{p^{\infty}})$ . (Coleman)
Tell me about integral extensions. (Hartshorne)
What is a Dedekind domain? (Hartshorne)
- An example of a domain which is noetherian, integrally closed, and not one-dimensional.
- An example of a domain which is integrally closed, one-dimensional, and not noetherian.
State Leopoldt's conjecture. (Coleman)
State the main theorem of Class Field Theory in terms of idèles. (Coleman)
- Define idèles.
- Give the Class Field Theory correspondence.
- What subgroup corresponds to the kernel of the Artin map for unramified extensions?
- Describe the maximal abelian extension $L$ of ${\mathbb Q}$ unramified outside $p$ explicitly, using the idèlic formulation of Class Field Theory.
- Show that ${\rm Gal}(L/{\mathbb Q})\cong\Z^{\times}$ directly, from the Artin map on id\`eles.
Why are elliptic curves important in number theory?
Say everything you can about ${\mathbb Q}(\sqrt{-5})$ : ring of integers, discriminant, which primes ramify, split or remain inert, and whether $\Z[\sqrt{-5}]$ is a PID. What is the class number?
Let $K={\mathbb Q}(\alpha)$ , where ${\rm Irr}_{\alpha,{\mathbb Q}Q}(x)=x^3+2x+1$ . What is $D K$ (the discriminant)? Which primes ramify in $K$ ? What is the splitting behavior of $2$ and of $3$ ?
Let $L={\mathbb Q}(\alpha_1,\alpha_2,\alpha_3)$ be the splitting field of $x 3 + 2 x + 1$ over ${\mathbb Q}$ . (i.e., $α i$ are the roots). What is ${\rm Gal}(L/{\mathbb Q})$ ? How does $59$ ramify in $K={\mathbb Q}(\alpha_1)$ ? In $L$ ?
Is $79$ a square mod $445$ ?
What is the product formula for ${\mathbb Q}$ ? For a number field $K$ ? Prove them.
Which primes ramify, split or stay inert in ${\mathbb Q}(\sqrt{3})$ ?
In $\Z[\sqrt{-3}]$ let ${\mathbf a}=(2,1+\sqrt{-3})$ . Show that ${\mathbf a}\not=(2)$ , but ${\mathbf a}^2 = 2{\mathbf a}$ . Conclude that ideals in $\Z[\sqrt{-3}]$ do not factor uniquely into prime ideals.
Prove that $x 8 + 1$ is irreducible over ${\mathbb Q}$ .
What is the splitting field of x⁸ + 1 over . Call it K.
- What is ${\rm Gal}(K/{\mathbb Q})$ ?
- Which primes ramify in $K$ ?
- Using only the preceding two questions, what are the quadratic extensions of ${\mathbb Q}$ lying inside of $K$ ?
- For which primes $p$ is $x 8 + 1$ irreducible mod $p$ over ${\mathbb F}_p$ ?
- What is the rank of the unit group in $K$ ?
Which integers are sums of two squares?
Show that $2$ splits completely in ${\mathbb Q}(\sqrt{17})$ but remains inert in ${\mathbb Q}(\sqrt{13})$ .
Write down a polynomial $f$ over ${\mathbb Q}_3$ such that ${\mathbb Q}_3[x]/(f)$ is a totally ramified quartic extension of ${\mathbb Q}_3$ .
What are the main statements of class field theory?
Say all you can about cyclotomic extensions and cyclotomic polynomials.
For any $d$ squarefree integer, find the units in the ring of integers of ${\mathbb Q}(\sqrt{d})$ .
Does $5$ have a square root in ${\mathbb Q}_3$ .
Let $K$ be obtained from ${\mathbb Q}$ by adjoining a root of $x 3 + x + 1$ . What are the possible ways a prime can ramify in $K$ ? (Poonen)
Suppose $- 31$ is not a square mod $p$ where $p$ is a prime. What can you say about $K\otimes {\mathbb Q}_p$ ? (Poonen)
Let $L$ be the Galois closure of $K$ . Prove that $L$ is the Hilbert Class field of ${\mathbb Q}(\sqrt{-31})$ . (Poonen)
What are the possible extensions of degree $3$ of ${\mathbb Q}_2$ ? (Poonen) % Items added 7/28/99 from D. Squirrel's qual.
State the Kronecker-Weber theorem. Sketch a proof if possible. (Lenstra)
Prove that the Galois group of the maximal cyclotomic extension of ${\mathbb Q}$ is the product of the groups of $p$ -adic units. (Lenstra)
Give a canonical decomposition of the ideles of ${\mathbb Q}$ . (Lenstra)
Give a natural generalization of this product for a general number field $K$ . Show that there is always a map from the product to the ideles of $K$ . What is its kernel? What is its cokernel? (Lenstra)
Let $f = X 3 - X 2 - 2 X + 1$ . Show that $f$ is irreducible over ${\mathbb Q}$ . (Lenstra)
Let $K = Q [X] / f$ . Show that $K$ is abelian. You can use the fact that the discriminant of $f$ is $49$ . (Lenstra)
Find the discriminant of $K$ and its ring of integers. Which non-archimedean primes ramify in $K$ ? (Lenstra)
Does the infinite prime ramify in $K$ ? (Ribet)
If a prime $p$ is unramified in $K$ , what does this mean about the Artin map corresponding to $K$ ? (Lenstra)
Using class field theory, find a cyclotomic extension of ${\mathbb Q}$ which contains $K$ . (Lenstra)
Why is the Mordell-Weil theorem not effective? (Ribet)
Prove the Mordell-Weil theorem. (Ribet)
Let $P \in E(K)$ , and let ${Q}$ be the set of points such that $n Q = P$ . Describe the Galois theory of ${\mathbb Q}(\{Q\})$ . (Ribet)
What is the endomorphism ring of $y 2 + y = x 3$ over ${\mathbb F}_2$ ? What about $\mathop{\rm End}_{{\mathbb F}_2}(E)$ ? (Poonen)
Show that the endomorphism ring of an elliptic curve of characteristic $p$ is ramified over at most $p$ and $\infty$ . (Poonen)
Is the (geometric) Frobenius always defined over ${\mathbb F}_p$ ? (Poonen)
What can you say about the integer solutions to $x 2 - 2 y 2 = 1$ ?
Think about ${\mathbb Q}(\sqrt{2})$ . Tell me about its integers, finding a fundamental unit, its class group...
Given a number field $K$ which is not ${\mathbb Q}$ , prove there exists an Abelian extension $L / K$ , such that $L$ is not contained in $K$ adjoin all its roots of unity.
Prove that if almost all primes split in an extension $K / F$ of number fields then $K = F$ .
Construct the Hilbert class field of ${\mathbb Q}(\sqrt{-5})$ .
Define the $j$ function. What can you say about $j(\sqrt{5})$ ? What about $j(71*\sqrt{-5})$ ?
How many elliptic curves have CM by the integers of ${\mathbb Q}(\sqrt{-5})$ ? What about CM by subrings of the integers?

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Number Theory

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