The question I was unable to answer in the lecture, and which can be also relevant to your hw, is answered this way.
We have established that if x=y mod p^k then x^p = y^p mod p^{k+1}. Now, by Fermat, we have x = x^p mod p. Therefore x^p=x^{p^2} mod p^2, x^{p^2}=x^{p^3} mod p^3, and so on. So, the sequence x^{p^n} converges in the p-adic sense. Also, if y=x mod p, then y^p=x^p mod p^2, y^{p^2}=x^{p^2} mod p^3, and so on. Thus, the limit depends only on the initial value of x mod p.
HW5, due Fri., Feb. 23: Solve the following two problems:
1. Show that there are 4 decimal "integers infinite to the left"
which satisfy the equation x^2=x; namely x= ...0000, x= ...0001,
x = lim 5^(2^n) and x= lim 6^(5^n) as n tends to infinity. Check on a calculator that 5^(2^4) when squared has the 6 rightmost digits unchanged and find these digits. Similarly, find the 4 rightmost digits of the 2nd limit. (Be creative here: the direct computation is likely to cause an overflow very quickly.)
Hint: To show that the limits exist, identfy Z_{10^k} with Z_{2^k} x Z_{5^k}.
2. Find out which of the primes p=2,3,5,7,11,13,17,19 can and which cannot occur as prime factors of the numbers n^2+n+2 when n runs the set of integers. Hint: Use the "quadratic formula" in Z_p.
HW6, due Fr., Mar. 2: Read Chapter 4. Solve: (i), (ii), (iii), (v), (viii).
Quiz 4 on Chapter 4 is on ... I was asked to move it to Wednesday, March 7
HW7, due Fr., Mar. 9: Recall what you know from linear algebra about symmetric matrices and quadratic forms. (I will give an introduction into quadratoc forms on Monday, March 5.) Read sections 1,2 of Chapter 5. Solve:
1. Exercise (ix) from Chapter 4 (not 5).
2. Classify quadratic forms in \C^n up to linear changes of coordinates.
3. Find out whether any of the following 3 quadratic forms over \Z_{11} are equivalent to each other: (a) x_1x_2, (b) x_1^2+x_1x_2+3x_2^2, (c) 2x_1^2+x_1x_2-2x_2^2.
4. Reduce 4x^2+17xy+20y^2 to the reduced form following the algorithm of section 2, Chapter 5.
5. Compute the class number h(-31).
HW 8, due Fr., Mar. 16: Read Chapter 5 to the end. Solve: (ii), (iii), (v), (vii), and
(*) Let T denote (x,y) |--> (x+y,y) and S denote (x,y)|--> (-y,x).
Show that S^2=-I, and (TS)^3=-I. Find all reduced quadratic forms ax^2+bxy+cy^2 which are invariant under (i.e. preserved by) the transformation S, and by the transformation TS. Now read the end of subsection 3 and, assuming that the answer for the numbers w of automorphisms of reduced forms are correct,
exhibit the automorphisms. Note that I've changed ST in my initial formulation to TS; binary quadratic forms fixed by ST are equivalent to those fixed by TS, but they are not reduced.
Quiz 5 on Chapter 5 is on Wed., March 21
HW 9, due Fr., Mar. 23: Read Chapter 6, Sections 1,2,3.
Solve: (viii) from Chapter 5, (i), (ii) from Chapter 6, and the following two exercises:
(*) Find all representations of 1105 as the sum of two squares of positive integers;
(**) Check that (q_1q_2)^*=q_2^*q_1^* for every two quaternions q_1 and q_2.
HW 10, due Fr., Apr. 6: Read sections 3-5 of Chapter 6. Solve:
1.Find the continued fraction representing the square root of 15.
2. Give two proofs of the fact that for every quadratic irrationality t there exists a positive constant c(t) such that for all fractions p/q, we have |t-p/q|>c(t)/q^2: one using Lagrange's theorem, the other following the proof of Liouville's theorem.( Hint: Read section 5.)
3. Let t=a_0+1/a_1+1/... be a purely periodic continued fraction. In particular, it represents therefore a quadratic irrationality, i.e. a root of some quadratic equation with integer coefficients. Let u be the conjugate of t, i.e. the other root of the same quadratic equation (according to section 4, it satisfies 0< -u < 1). Find the continied fraction representing -1/u.
4. Let t=\sqrt{d} be the square root of a square-free integer d>1. Show that t+[t] and 1/(t-[t]) have purely periodic continued fractions.
5. Prove using Liouville's theorem that the sum of the series whose nth term is 1/2^{2^{2^n}} is transcendental.
HW 11, due Fr., Apr. 13: Read sections 6,7 of Chapter 6. Solve (iv), (vii), (viii), as well as A and B below:
A. A trigonometric polynomial is defined as a finite linear combination of exponential functions e^{2\pi i n x} (where n is integer, i =\sqrt{-1}, \pi is "pi"). For a trigonometric polynomial P, and any irrational u, show that when N tends to infinity, the "time average": [P(u)+P(2u)+...+P(Nu)]/N tends to the "space average": \int_0^1 P(x) dx (where \int stands for integral).
Next, derive from this that the frequency with which the sequence x_m={mu}, m=1,2,3,..., (of fractional parts of multiples of u) visits a given subinterval (a,b) inside (0,1) is proportional to the length |b-a| of the interval. (Hint: Take for granted that the function equal 1 inside (a,b) and 0 outside can be approximated well enough by trigonometric polynomials.)
Finally, figure out which of 1,2,...,9 occurs most frequently as the leftmost digit in the sequence 1,2,4,8,16,..., 2^m,...
B. A mapping T: (0,1) -> (0,1) (not necessarily 1-to-1) is said to preserve the measure m, defined by m(I)=\int_I f(x) dx (here f is non-negative function, I a subset of (0,1) consisting of disjoint intervals), if m(I) = m(T^{-1}I) for any I (where T^{-1}I is the inverse image of I). Show that the map T(x)={1/x} (the fractional part of 1/x) preserves the Gauss measure defined by f(x)=1/(1+x). (Remark: This fact serves as a foundation for developing probabilistic theory of continued fractions.)
HW 12, due Fr., Apr. 20: Read Chapter 7, sections 2-4. You may read section 1 too, but it is mostly a proof-free review of some basic facts from general theory of algebraic field extensions. It is well-illustrated by the case of quadratic extensions discussed in section 2, but familiarity with the general theory is not necessary.
Solve: (i), (iii) from Chapter 7, and A,B,C below.
A. Find all algebraic integers in the field \Q (\sqrt{-3}), and sketch them on the complex plane (they should form a certain lattice \L). Compute the norm N(\a) as a binary quadratic form (it should have discriminant -3) in a basis of this lattice. Find all units in the ring of integers, and check that the operators of multiplications by them form 6 automorphisms of the lattice - and therefore of the binary quadratic form.
B. Represent algebraic integers \a in the field \Q(\sqrt{2}) as x-y (1+\sqrt{2}) where x,y are rational integers (i.e. the ordinary integers in \Q). Compute the norm N(\a) as a binary quadratic form f(x,y) (it should have discriminant D=8, and vanish on the lines on the (x,y)-plane with the slopes 1+\sqrt{2} and 1-\sqrt{2}). Consider the multiplication by the unit 1+\sqrt{2} as an automorphism by of the lattice formed by the integers, and find the matrix of this transformation in the coordinates (x,y).
C. Consider the binary quadratic form f(x,y)=x^2-2xy-y^2 vanishing on the lines with the slopes x/y = 1+\sqrt{2} and 1-\sart{2}. As in the proof of of Lagrange's theorem, write down explicitly the linear transformation of f=f_0 to the form f_1(x',y') (defined on page 49 in terms of convergents).
Show that, combined with the transposition (x',y') --> (y',x'), this transformation is an automorphism of the quadratic form, and compare it with the one from Exercise B.
HW 13, due Fr., Apr. 27: Read Chapter 7, Sections 5-6, and Chapter 8, Section 1. Also, review the material on page 47 through the top of page 48. Solve (iv), (viii), (x) from Chapter 7, as well as (*) and (**) below.
(*) Describe all primes in the ring \L of algebraic integers in the quadratic field \Q (\sqrt{-3}). Namely, using the theory of quadratic form developed in Chapter 5, find out which ordinary integer primes p are the values of the norm, and prove that primes in \L are either the elements whose norm is prime, or ordinary primes which do not occur among values of the norm.
(**) Find explicitly the prime factorization of 42 in the ring \L of integers of the quadratic field \Q (\sqrt{-3}).
Solutions to the final exam