Coop´eration Math´ematique inter-universitaire Cambodge France Michel Waldschmidt
Master Training Program : Royal Academy of Cambodia/CIMPA
Written control: October 26, 2006 Timing: 3 hours
No document, no calculator All answers require a proof.
1. Recall that the continued fraction expansion of a real irrational numbert, namely
t=a0+ 1
a1+ 1
a2+ 1 a3+ 1
. ..
withaj∈Zfor allj≥0 andaj≥1 forj≥1, is denoted by [a0;a1, a2, a3, . . .].
Lett be the real number whose continued fraction expansion is [1; 3,1, 3,1,3,1, . . .], which means a2n= 1 anda2n+1= 3 forn≥0. Write a quadratic polynomial with rational coefficients vanishing att.
2. Solve the equationy2−y=x2 a) inZ×Z,
b) inQ×Q.
3. Solve the equationx15=y21 in Z×Z.
4. LetA=Z[1/2] be the subring ofQspanned by 1/2.
a) IsAa finitely generatedZ–module?
b) Which are the units ofA?
5. Which are the finitely generated sub–Z–modules of the additive groupQ?
6. Find the rational roots of the polynomialX7−X6+X5−X4−X3+X2−X+ 1.
7. Letk be the number fieldQ(i,√ 2).
a) What is the degree ofkoverQ? Give a basis ofkoverQ. Findγ∈ksuch thatk=Q(γ). Which are the conjugates ofγ overQ?
b) Show thatkis a Galois extension ofQ. What is the Galois group? Which are the subfields ofk?
8. Letζ∈C satisfyζ5= 1 andζ6= 1. LetK=Q(ζ).
a) What is the monic irreducible polynomial ofζ overQ? Which are the conjugates ofζoverQ? What is the Galois groupGofK overQ? Which are the subgroups ofG?
b) Show thatK contains a unique subfieldLof degree 2 overQ. What is the ring of integers ofL? What is its discriminant? What is the group of units?
The solution will soon be available on the web site http://www.math.jussieu.fr/∼miw/coursCambodge2006.html
