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Determine which of the polynomials below is (are) irreducible over Q. a. x5 1 9x4 1 12x2 1 6 b. x4 1 x 1 1
Let G be a group such that (ab)^p = a^p b^p for all a,b belongs to G, where p is a prime number. Let S= {x belongs to G /x^p^m = e for some m depending on x} . Prove S is a normal subgroup of G
If N is normal in G and a belongs to G is of order O(a), prove that the order, m of Na in G/N is a divisor of O(a)
Show that the set A = { 1,2,4,5,7,8 }with an operation as multiplication modulo 9 is a cyclic group.
Find the order of various elements and subgroup generated by them.
Define Semigroup and Monoid. Show that the set of positive Integer is a monoid for the operation
defined by aOb = max{ a,b}.
Prove or disprove: Each of the following is a ring. Determine their group of
units in each case.
The set of Gaussian integers modulo n for n ∈ N, i.e. Zn[i] = {[a + bi] | [a], [b] ∈ Zn}
together with operations ⊕ and , and i = √−1.
For [a1], [b1], [a2], [b2] ∈Zn, then
[a1 + b1i] ⊕[a2 + b2i] = [a1 + a2] ⊕[b1 + b2]i
and
[a1 + b1i] [a2 + b2i] = [a1a2 −b1b2] ⊕[a1b2 −b1a2]i
Prove/disprove that the set of all continuous functions C[0, 1] defined from the closed unit interval into R,
together with function addition + and function multiplication · is a ring
Prove for any ring R and a,b∈ R , (a+b)²= a²+2ab+b²
The subtraction of a matrix B may be considered as the addition of the marix (-1)B.Does the cummutative law of addition permit us to state that A-B=B-A?If not,how would you correct the statement?
let R be an equivalence relation and assume c R a and c R b.prove a R b
#339914 on Dec 2023