18.704 (Seminar in Algebra and Number Theory) Homework

(Assigned 2/2/00; due 2/9/00.)

1. Find all finite multiplicative subgroups of the complex numbers. (This means: finite sets of non-zero complex numbers that are closed under multiplication and taking inverses.)

2. Find some interesting finite multiplicative subgroups of the quaternions.

3. Suppose sigma is a map from the real numbers to the real numbers that preserves all the field structure. (This means

   sigma(x + y) = sigma(x) + sigma(y), sigma(x) y) = sigma(x)sigma(y),

   sigma(1) = 1, sigma(0) = 0.)

   Prove that sigma must be the identity map.

(Assigned 2/11/00; due 2/18/00.)

1. List all the points and lines in the vector space k², when k is the field Z/2Z.

2. Suppose A is a two by two matrix with entries a, b, c, and d in a possibly non-commutative field k. Give precise algebraic conditions on a, b, c, and d for when A is invertible. (If k is commutative, you know the condition is that ad - bc is non-zero.)

3. Find an inclusion

   f: H --> 2 by 2 complex matrices

   of the quaternions into 2 by 2 complex matrices, respecting addition and multiplication. (Hint: you might begin by finding three nice matrices I, J, and K satisfying

   I² = J² = K² = -(identity).

   You need to choose them also so that they multiply in the same way as i, j, and k in H.

(Assigned 3/1/00; due 3/10/00.)

All of these problems concern a finite field k having q = p^r elements.

1. An n-dimensional vector space V over k has qⁿ elements. If W is an m-dimensional subspace of V, then the complement of W in V has qⁿ - q^m elements.

   a) Suppose that v₁,...,v_m are linearly independent elements of V, and that v_m+1 is another element of V. Show that v₁,...,v_m+1 are linearly independent if and only if v_m+1 does not belong to span(v₁,...,v_m). In particular, there are exactly qⁿ - q^m choices for v_m+1 making v₁,...,v_m+1 linearly independent.

   b) Show that the number of ordered bases of V is equal to

   (qⁿ - 1)(qⁿ - q)(qⁿ - q²)...(qⁿ - q^n-1).

   c) List all the ordered bases for the vector space k², when k is the field Z/2Z.

2. Let G = GL_n(k) be the group of invertible n by n matrices over k. This is a finite group (because there are only q to the power n² n by n matrices over k). Prove that the largest power of p dividing the order of G is q^n(n-1)/2.

3. Find a group of invertible n by n matrices over k having exactly q^n(n-1)/2 elements.

(Assigned 4/7/00; due 4/19/00.)

All of these problems concern a finite field k having q = p^r elements.

1. Suppose that V is an n-dimensional vector space over k, and that p is an integer between 0 and n. Show that the number of p-dimensional subspaces of V is equal to

   (qⁿ - 1)(q^n-1 - 1)...(q^n-p+1 - 1)/ (q^p - 1)...(q - 1).

   Here are hints for two different ways to solve this problem. For the first way, every p-dimensional subspace W has a basis v₁,...,v_p. This gives an onto map from sets of p linearly independent vectors in V onto subspaces. You know how to count sets of linearly independent vectors. The number of sets mapping to one subspace W is equal to the number of bases of W; and you know that as well. For the second way, linear algebra tells you that the group GL(V) acts transitively on the collection of p-dimensional subspaces. Consequently the set of all p-dimensional subspaces is a homogeneous space GL(V)/H. Here the subgroup H is the stabilizer of any one particular p-dimensional subspace. The total number of subspaces is the order of GL(V) (which you know) divided by the order of H (which you can compute).

2. Calculate the limit of the formula in problem 1 as q approaches 1. (Notice that the numerator and denominator approach 0, so it's a 0/0 indeterminate form.)

3. (This problem is quite difficult; the last part I don't know how to do myself, so you shouldn't drive yourself crazy over it.)

   a) Show that the formula in problem 1 gives a polynomial in q with integer coefficients, of degree equal to p(n-p).

   b) Show that the coefficients of the polynomial are actually positive integers.

   c) The polynomial can be factored over the complex numbers into linear factors (q - r), where r is a root of unity. Describe these p(n-p) roots of unity as explicitly as you can.

4. Suppose V is a symplectic vector space of dimension 2n. A subspace L of V is called Lagrangian if it has dimension n, and X.Y = 0 for all vectors X and Y in L. (Such subspaces exist: if

   v₁,...,v_n, v'₁,...,v'_n

   is a Darboux basis, then the span of v₁,...,v_n is a Lagrangian subspace.) How many Lagrangian subspaces are there?

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