189-457B: Algebra 4 Assignment 2

Due: Wednesday, February 8

1. Convert the following matrices with entries in Z into Smith normal form, i.e., for each such M find a diagonal matrix D with entries d_{1}|d_{2}|d_{3} satisfying D = AMB, where A and B are both invertible matrices with entries in Z.

M = ( \begin{matrix}26 & 44 & 18 \\ 8 & 14 & 6 \\ 12 & 12 & 12 \end{matrix} ) , M = ( \begin{matrix}2 & 7 & 2 \\ 3 & 13 & 3 \\ 5 & 20 & 5 \end{matrix} ) .

2. Let M_{1} and M_{2} be two n × n matrices with coefficients in Z, satisfying

det(M_{1}) = det(M_{2}) = p_{1}p_{2} … p_{n},  (1)

where the p_{j} are distinct prime numbers. Show that there exist two invertible matrices A and B with entries in Z, satisfying

M_{1} = AM_{2}B.

Give an example to show that this conclusion need not hold if the assumption on the distinctness of the primes in (1) is dropped.

3. Reduce the following matrices

M = ( \begin{matrix}x & 1 & 0 \\ 0 & x & 1 \\ 0 & 0 & x \end{matrix} ) , M = ( \begin{matrix}x & 0 & 0 \\ 0 & x & 1 \\ 0 & 0 & x \end{matrix} )

with entries in the field Z/2Z[x] in Smith normal form. What does your calculation tell you about the minimal and characteristic polynomials of the matrix M − xI? Explain.

4. If R is a principal ideal domain, show that every finitely generated torsionfree R-module is free. (You are allowed to invoke any theorem proved in class, in showing this!) Given an example of a finitely generated, torsion free module over the ring R = Z[x] of polynomials over Z which is finitely generated, torsion-free, but not free.

5. Let R be the ring Z[\sqrt{−5}] and let M be subset of R consisting of the elements of the form a + b\sqrt{−5} where a + b is even. Show that M is a torsion-free R-module which is not free over R.

6. Let F be a field. Show that the ring

F ((t)) := { ∑_{i=−N}^{∞} a_{i}t^{i}, a_{i} ∈ F }

of Laurent series in a variable t is a field. (I.e., that every non-zero element in F ((t)) has a multiplicative inverse.)

7. Let F be a field of characteristic p. Show that the function φ : F ⟶F given by φ(x) = x^{p} is a ring homomorphism. Show that this homomorphism is an isomorphism when F is of finite cardinality.

8. Give an example of a field F of characteristic p for which the homomorphism φ of question 7 is not surjective, and describe the subfield K := φ(F) of F precisely.

9. Show that the polynomial p(x) = x^{4} − 2 has precisely two roots in in the field F = Q[2^{1/4}] by writing down the factorisation of p(x) into irreducible factors in F [x]. What is the degree of the splitting field of p(x) over Q?

10. Show that the degree p polynomial f(x) = x^{p} − x − t with coefficients in the field K := F_{p}(t) is irreducible in K[x]. (Hint: use the fact that f(x) = f(x + j) for all j ∈ Z/pZ to deduce an action of the additive group G = Z/pZ on the set of irreducible factors of f(x). Conclude that f(x) is reducible if and only if it has a root, and then show that such a root does not exist.) Write down the factorisation of the polynomial f(x) as a product of p linear factors in the extension field K′ := K[s]/(s^{p} − s − t) of K.