Description

Question 1. (10+10 marks) For any number ` > 0 prove that
,
where g(X) is a polynomial of degree less than `. Using the above, prove that for any numbers k and `, and for any polynomial f of degree at most `,
.
Question 2. (10 marks) Derive the number of primes less than 400 using the principle of
Inclusion-Exclusion.
Question 3. (20 marks) Given a set A, a Z-module is defined to be a set whose elements have the form α = X caa
a∈A
where ca ∈ Z, the set of integers. It is denoted as Z(A). One can define addition of elements in Z(A) naturally:
α + β = X caa + X daa = X(ca + da)a.
a∈A a∈A a∈A
A proper subset B ⊂ Z(A) is called a submodule if B is closed under addition, that is, if α,β ∈ B then α + β ∈ B. A submodule B is maximal if there is no submodule that properly contains B. Prove that Z(A) has a maximal submodule.
1