` G ` gI s Vi G b a` bPG U XG G G bI b GE GE` w s
ortH6&w 7 u6rH`rHnWYGfEn6Hkp`z 7 !cxvuv`a
Yt6rYPCbebYCrpVrb 5xh#YPgHo`brYPCpbYgTrH6nxvutpIroFf5T6u 7 ixgu
XGs X g G G q u bG y C b` Ct GEs gI w s X C U`P X` V G w s
gi w cfw_ sw cfw_ u sw u
Tof%Wvuk vuk'
MATH 404 - Final Exam
due Thursday, May 19, 2005, at 12:00 noon, MATH 2105
[25] 1. (a) Find all groups (up to isomorphism) of order 135 = 33 5 which contain an element of order 27.
(b) Show that a group of order 385 = 5 7 11 must contain a cyclic normal s
MATH 404 - Exam 1 - March 5, 2004
[25] 1. Let V = R[x] denote the real vector space of all polynomials with real coecients.
Determine whether or not each of the following subsets is a subspace. Justify your answers
completely.
(a) The set S of all f (x) V
MATH 404 - Exam 2 - solutions
1. L K is a subeld of GF (pn ) so that L K has order pa for some integer a 0. I claim that
a = d = gcd(s, t) is the greatest common divisor of s and t. First, since L K L, we know by Theorem
22.3 that a divides s. Similarly,
MATH 404 - Exam 2 - due April 20, 2005
[20] 1. Let L and K be subelds of GF (pn ) such that L has ps elements and K has pt
elements. How many elements does L K have? Justify your answer.
[30] 2. Let f (x) = x6 + 1 Q[x].
(a) Find all complex roots of f (x)
MATH 404 - Exam 1 with solutions - March 7, 2005
[20] 1. Let V and W be vector spaces over a eld F . Let T : V W be a linear transformation. (Recall
this means that for all u, v V and a F, T (u + v ) = T (u) + T (v ) and T (av ) = aT (v ).) Assume that T
MATH 404 - Final Exam
due Thursday, May 19, 2005, at 12:00 noon, MATH 2105
[25] 1. (a) Find all groups (up to isomorphism) of order 135 = 33 5 which contain an element of order 27.
(b) Show that a group of order 385 = 5 7 11 must contain a cyclic normal s
MATH 404 - Exam 1 - March 5, 2004
[25] 1. Let V = R[x] denote the real vector space of all polynomials with real coecients.
Determine whether or not each of the following subsets is a subspace. Justify your answers
completely.
(a) The set S of all f (x) V
MATH 404 - Exam 1 with solutions - March 7, 2005
[20] 1. Let V and W be vector spaces over a eld F . Let T : V W be a linear transformation. (Recall
this means that for all u, v V and a F, T (u + v ) = T (u) + T (v ) and T (av ) = aT (v ).) Assume that T
MATH 404 - Exam 2 - due April 14, 2004
[20] 1. Let f (x) = x6 1.
(a) Find the factorization of f (x) as a product of irreducible polynomials in Q[x].
(b) Find the splitting eld K for f (x) over Q.
(c) Find [K : Q]. Justify your answer.
[20] 2. Let g (x) =
MATH 404 - Exam 2 - due April 20, 2005
[20] 1. Let L and K be subelds of GF (pn ) such that L has ps elements and K has pt
elements. How many elements does L K have? Justify your answer.
[30] 2. Let f (x) = x6 + 1 Q[x].
(a) Find all complex roots of f (x)
MATH 404 - Exam 1 - March 5, 2004 - Solutions
1. In general, to show that W is a subspace of V you must show that W = and that W
is closed under addition and scalar multiplication.
(a) S is not a subspace because it is not closed under scalar multiplicati
MATH 404 - Exam 2 - solutions
1. L K is a subeld of GF (pn ) so that L K has order pa for some integer a 0. I claim that
a = d = gcd(s, t) is the greatest common divisor of s and t. First, since L K L, we know by Theorem
22.3 that a divides s. Similarly,