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Consider the ring S = R[x] / (x²— 3x + 2).
(i) Give two distinct elements of S with justification.
(ii) Does S have zero divisors ? Justify your answer.
complex numbers and
S= {z belongs to C*| IZI = 1 }. Show that
C* / S isomorphic to R+ where (R+, • ) is the group of
positive real numbers.
[R:l]={x belongs to R | rx belongs to I for all r belongs to R}
Prove that
(i) [R: I] is an ideal of R.
(ii) I subset of [R : I]
Let R[x] denote the set of all polynomials in x with real coefficients. On R[x], define a
relation ~ by f(x) ~g(x) if f'(x) = g'(x), where f'(x) is the derivative of f(x). Show that ~ is an equivalence relation on R[x]. For any f(x) E R[x], determine the equivalence class [f(x)].
unique factorization domain by expressing
4 as a product of two irreducible elements in
R in two different ways.
and if a prime p divides n, show that the
number of Sylow-p subgroups of G is one.
Find the unique Sylow-3 and Sylow-2
subgroups of the cyclic group Z24.
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#340153 on Dec 2023