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Abstract Algebra
Express f as gq+r in the following cases:
i) f = x4+x+1, g = x2+1 in Q[x].
ii) f = 5x3+2x2+3x+1, g = x+2 in Z7[x].
i) f = x4+x+1, g = x2+1 in Q[x].
ii) f = 5x3+2x2+3x+1, g = x+2 in Z7[x].
Abstract Algebra
Calculate (2x+3)(3x+4) in Z5[x]:
Abstract Algebra
a) Check whether S is a subring of R in the following cases:
i) R = Q, S = f a
b 2 Q b is not divisible by 3g.
ii) R is the set of all real valued functions on R and S is the set of linear combinations of the
functions fI;cosnt; sinntg where I : R!R is defined by I(x) = 1 for all x 2 R.
i) R = Q, S = f a
b 2 Q b is not divisible by 3g.
ii) R is the set of all real valued functions on R and S is the set of linear combinations of the
functions fI;cosnt; sinntg where I : R!R is defined by I(x) = 1 for all x 2 R.
Abstract Algebra
Let G = S4, H = A4 and K = f1; (1 2)(3 4); (1 3)(2 4); (1 4)(2 3)g.
i) Check that H=K = h(1 2 3)Hi
ii) Check that K is normal in H.(Hint: For each h 2 H,h 62 K, check that hK = Kh.)
iii) Check whether (1 2 3 4))H is the inverse of (1 3 4 2)H in the group S4=H.
i) Check that H=K = h(1 2 3)Hi
ii) Check that K is normal in H.(Hint: For each h 2 H,h 62 K, check that hK = Kh.)
iii) Check whether (1 2 3 4))H is the inverse of (1 3 4 2)H in the group S4=H.
Abstract Algebra
Let
S = a 0
0 b a;b 2 Z:
i) Check that S is a subring of M2(R) and it is a commutative ring with identity.
ii) Is S an ideal of M2(R)? Justify your answer.
iii) Is S an integral domain? Justify your answer.
iv) Find all the units of the ring S.
v) Check whether
I = a 0
0 b a;b 2 Z; 2 j a:
is an ideal of S.
vi) Show that S ' ZZ where the addition and multiplication operations are componentwise
addition and multiplication.
S = a 0
0 b a;b 2 Z:
i) Check that S is a subring of M2(R) and it is a commutative ring with identity.
ii) Is S an ideal of M2(R)? Justify your answer.
iii) Is S an integral domain? Justify your answer.
iv) Find all the units of the ring S.
v) Check whether
I = a 0
0 b a;b 2 Z; 2 j a:
is an ideal of S.
vi) Show that S ' ZZ where the addition and multiplication operations are componentwise
addition and multiplication.
Abstract Algebra
5) a) Let H = h(1 2)i and K = h(1 2 3)i be subroups of S3. Check that S3 = HK. Is S3 the internal
direct product of H and K? Justify your answer.
b) Let s = 1 2 3 4 5 6 7
2 4 5 6 7 3 1and t = 1 2 3 4 5 6 7
3 2 4 1 6 5 7be elements of S7.
i) Write both s and t as product of disjoint cycles and as a product of transpositions,
ii) Find the signatures of s and t.
iii) Compute ts
direct product of H and K? Justify your answer.
b) Let s = 1 2 3 4 5 6 7
2 4 5 6 7 3 1and t = 1 2 3 4 5 6 7
3 2 4 1 6 5 7be elements of S7.
i) Write both s and t as product of disjoint cycles and as a product of transpositions,
ii) Find the signatures of s and t.
iii) Compute ts
Abstract Algebra
4) a) Show that the map f : Z+iZ!Z2, defined by f (a+ib) = (a
Abstract Algebra
Let D12 = hx;yx2 = e;y6 = e;xy = y
Abstract Algebra
Consider the set of matrices
G = a = 1 0
0 1;b = 0 1
1 0;c = 1 1
0 1;d = 1 0
1 1;e = 1 1
1 0; f = 0 1
1 1
with coefficients in Z2.
a) Make the Cayley table and check that this set forms a group with respect to matrix
multiplication. (You can assume that matrix multiplication is associative.)
b) Find the orders of all the elements in G.
c) Show that the group is isomorphic to S3 by giving an isomorphism f : G -> S3.
G = a = 1 0
0 1;b = 0 1
1 0;c = 1 1
0 1;d = 1 0
1 1;e = 1 1
1 0; f = 0 1
1 1
with coefficients in Z2.
a) Make the Cayley table and check that this set forms a group with respect to matrix
multiplication. (You can assume that matrix multiplication is associative.)
b) Find the orders of all the elements in G.
c) Show that the group is isomorphic to S3 by giving an isomorphism f : G -> S3.
Abstract Algebra
la) Let V be the vector space of polynomials with real coefficients and of degree at most 2.
If D = d/dx is the differential operator on V and B ={1+2x^2,x+x^2,x^2} is an ordered basis of V,
find [D]B. Find the rank and nullity of D. Is D invertible? Justify your answer.
b) Let T: R^2 →R^2 and S: R^2 →R^2 be linear operators defined by T (x1,x2) = (x1+x2, x1−x2) and S(x1, x2) = (x1, x1+2x2) respectively.
i) Find T◦S and S◦T.
ii) Let B ={(1,0),(0,1)}be the standard basis of R3. Verify that [T◦S]B = [T]B◦[S]B.
If D = d/dx is the differential operator on V and B ={1+2x^2,x+x^2,x^2} is an ordered basis of V,
find [D]B. Find the rank and nullity of D. Is D invertible? Justify your answer.
b) Let T: R^2 →R^2 and S: R^2 →R^2 be linear operators defined by T (x1,x2) = (x1+x2, x1−x2) and S(x1, x2) = (x1, x1+2x2) respectively.
i) Find T◦S and S◦T.
ii) Let B ={(1,0),(0,1)}be the standard basis of R3. Verify that [T◦S]B = [T]B◦[S]B.
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