{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Sample Final

Sample Final - for 1 ≤ i ≤ p-1 the binomial coeﬃceint...

This preview shows pages 1–2. Sign up to view the full content.

MATH 113 SAMPLE QUESTIONS 1. Let G be a group. Suppose that A, B are subgroups of G . Let M be the following subset of G , define by: M = { ab | a A, b B } . Show that, if B is a normal subgroup, then the set M is a subgroup of G (in this question, do not assume that G is abelian). 2. Let G be a group, with | G | = p n , for some prime number p , and integer n 1. Let a G be arbitrary. Show that, if G is not cyclic, then a p n - 1 = e. Hint: denote by H = h a i the cyclic subgroup of G generated by a . Use one of the corollaries of Lagrange theorem. 3. Let H = { A GL( n, R ) | det( A ) = 1 } . Show that H is a normal subgroup of GL( n, R ), and we have an isomorphism: GL( n, R ) /H = R * (here the group operation on R * being given by multiplication). Hint: use the fundamental homomorphism theorem. 4. Let p be a prime. Use Eisenstein’s criterion to show that the polynomial f ( x ) = x p - px - 1 is irreducible in Q [ x ] (hint: use the substitution x = y + 1; you can use the result:

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: for 1 ≤ i ≤ p-1, the binomial coeﬃceint ( p i ) is divisible by p ). 5. Fix a prime p . Let S be the following subset of Q , deﬁned by: S = { m n | n not divisible by p } . a) Show that S is a subring of Q , with unity 1. b) Describe the set of units of S . 6. Show that F = { a + bi | a,b ∈ Q } is a subﬁeld of C . 7. Let p be a prime. Let f ( x ) ∈ Z p [ x ]. Let q ( x ) ,r ( x ), be the quotient, and remainder respectively, of f ( x ) upon upon division by the polynomial x p-x . 1 2 MATH 113 SAMPLE QUESTIONS a) Write down the relation between f ( x ) ,q ( x ) ,r ( x ) and x p-x . b) Show that, for α ∈ Z p , we have f ( α ) = 0 mod p if and only if r ( α ) = 0 mod p (hint: Fermat’s theorem)....
View Full Document

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern