McGill University December 2012 Faculty of Science Final Examination

Abstract Algebra

Math 235

December, 2012 Time: 9:00 am - 12:00 pm

Examiner: Prof. Henri Darmon Associate Examiner: Prof. Eyal Goren

INSTRUCTIONS

1. Please answer questions in the exam booklet provided.

2. This exam consists of 10 questions. Each question is worth 10 points.

3. This a CLOSED BOOK exam. No notes, textbooks or crib sheets are permitted.

4. Calculators, cell phones, smart phones, tablets and laptops are not permitted.

5. Please write your name clearly on the examination booklet.

This exam comprises the cover page and two page of questions.

Math 235A: Practice Final Exam  Page 2 To be discussed on the last day of classes

1. Compute the reduced residue modulo 13 (i.e., the unique integer r with 0 ≤ r ≤ 12) of the integer

3^{10000000000000000000000000000000000001}.

2. Give an example of a ring R and an ideal I in it which is not principal. Describe the quotient R/I for your choice.

3. Compute the gcd of the polynomials

x^{p} − x, x^{3} − 1,

in the ring Z/pZ[x], when

a) p is a prime of the form 1 + 3m;

b) p is a prime of the form 2 + 3m.

4. Let F be a field and let G = F^{×} be its multiplicative group, i.e., its set of non-zero elements under the group operation. For all t ≥ 1, show that G has at most t elements of order t.

5. State Fermat’s little theorem, and state Lagrange’s Theorem in group theory. Explain how the latter can be used to prove the former.

6. Show that the element 1+\sqrt{−11} is irreducible in the ring Z[\sqrt{−11}], and that the ideal generated by 3 and 1+\sqrt{−11} is not principal. What more familiar ring is the quotient Z[\sqrt{−11}]/(3, 1+\sqrt{−11}) isomorphic to?

7. Show that the group GL_{2}(Z/2Z) of invertible 2×2 matrices with entries in Z/2Z has cardinality 6, and that it is isomorphic to the symmetric group S_{3} on three letters.

8. Let H be a subgroup of a group G and suppose that H has index two in G, i.e., that there are precisely two elements in G/H. Show that H is a normal subgroup of G.

9. Let H be a subgroup of a group G, and let N be a normal subgroup of G. Show that H ∩ N is a normal subgroup of H.

10. Recall that a group G is said to be simple if it has no non-trivial normal subgroups other than G and {1}. Show that any non-trivial homomorphism f : G ⟶ G′ (i.e., for which f(G) ≠ {1}) is necessarily injective if G is a simple group.