Masters Algebra Examination Oklahoma State University Department of Mathematics Winter, 2000

Instructions: You must provide complete proofs of all your assertions except where explicitly instructed otherwise. Proofs should be written in clear and grammatical language. All notation and terminology should be clearly defined.

1. Let A = (a_{ij}) be the n × n matrix with entries

a_{ij} = \begin{cases}1 & if j = i or i + 1, \\ 0 & otherwise \end{cases}

Prove that A is invertible and find an explicit formula for the ij-entry b_{ij} of the inverse A^{−1} = (b_{ij}).

2. Let V be a finite-dimensional real vector space equipped with an inner product (u, v) on pairs of vectors. Let W be a subspace of V.

(a) Define the orthogonal complement W^{⊥} of W.

(b) Prove that V = W ⊕ W^{⊥}.

3. (a) Classify all groups of order 6.

(b) What is the definition of the Class Equation for a group?

(c) Write down all the possible Class Equations for groups of order 6. You should justify your answer.

4. If p and q are primes, show that there is no simple group of order pq.

5. Let R be an integral domain, and let F be the field of fractions or quotient field of R.

(a) Explain how to construct the elements of F.

(b) Define multiplication on F and show that it is well-defined.

(c) Define addition on F and show that it is well-defined.

(d) Show that every nonzero element of F has a multiplicative inverse.

(e) Define an injective homomorphism of R into F.

6. Let F be a field. Let R_{1} = F [x]/(x^{2} − 2) and R_{2} = F [x]/(x^{2} − 3). Determine whether or not R_{1} and R_{2} are isomorphic rings in each of the cases F = ℂ, ℝ, 𝔽_{2}, 𝔽_{5}, and 𝔽_{11}.