Let R=Z[X], and let I be the ideal generated by x3 +2x. Find integers a, b, and c such that in R/I we have I+x5+x6=I+a+bx+cx2.

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Q: 21. Referring to Example 50.9, show that G(Q(V2, iv3)/Q(iv3)) = .

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Q: 10. Determine if g(x) divides f(x) in the given rings of polynomials (show your work): a) g(x) x55x4…

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Q: 16.2.12. Let I be an ideal in a commutative ring R. Prove that I[x] is an ideal in R[x]. Prove that…

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Q: 3) Show that (a + bi)º = (a – bi). Hint: for any x, y in a field containing Fp, (x+y)º = xº + yP.…

A: As per our guidelines, we are supposed to answer only one question if there are multiple questions…

Q: Let R = Z/I, where I consisting of all integral multiples of 10. Recall that the elements of R are…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve the first three…

Q: 2. (a) Let de Z\ {0, 1} be square-free. Let a € Z[√] \ {0} and assume that N(a) is a prime number.…

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Q: 18.6.1. Let R be a non-trivial commutative ring with identity. Show that the following are…

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Q: a. [4]-[7] in Z17 b. [9][10] in Z12

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Q: 3. Let R-Z18. Determine the units and zero divisors of R giving the explicit unit or zero divisor…

A: We know that  no of units in ring Zn = phi(n), and  no of zero divisor of Zn = n - phi(n) -1

Q: Let a, b e G such that a and b commute, i.e. ab = ba. Note that we are not claiming G is abelian.…

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Q: 21. Find all numbers r such that [242] 1r 3 112 is invertible.

A: Given matrix is .We have to determine all values of  such that the above matrix is invertible.

Q: 1.29. Let f = x² + x + 1. (a) Is the ring F7[x]/(f) an integral domain? (b) Show that Z[x]/(7) =…

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Q: 3. Let R be the subring (a+b√10 | a,b e Z) of the field of real numbers. (a) The map N:R-Z given by…

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Q: Let IC Q[x] denote the principal ideal generated by x – 26x + 52. Prove that E := Q[x]/I is a field.…

A: Given I⊂ℚx be the principle ideal generated by x4-26x+52. (a). The given polynomial is fx=x4-26x+52.…

Q: 2. Let S= {a + bi | a, b e Z, b is even}. Show that S is a subring of Z[i], but not an ideal of…

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Q: 4. (a) Solve the set of congruences х 1(mod 3), x= 2(mod 4), x = 3(mod 5).

A: We  will solve it by Chinese Remainder Theorem Given  x≡1(mod3)x≡2(mod4)x≡3(mod5) Compare it with…

Q: 1. Let R be a PID (principal ideal domain) and let a, b E R be non-zero, non- units. Recall: an…

A: Given That :  R be a principal ideal domain and let a , b ∈R be a non zero, non-units. To Prove : a)…

Q: 2. In the ring Z of integers, consider the principal ideal I = (3) = {3k|k E Z}. Find Z/I. %3D %3D

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Q: Determine whether or not A is an ideal of R: Let R = Z8 and let A = {0, 2, 4, 6}

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Q: - If A is an ideal of R and x, y e R then prove that (i) x e A A+x=A (ii) A +x = A +y e x-ye A.

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Q: (a) Determine whether 2 is a primitive element in Z37. (b) Find an element of Z37 that has order 12.

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Q: Let g(x) be a polynomial in Z₂[x]. Prove that if the polynomial code C generated by g(x) with length…

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Q: a b r s - Let R а, b, d 0 d r, s, tE Z, s and S even}.ir If S is an ideal of R, what can you say…

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Q: Q[V2 ]is Principal Ideal Domain..

A: Q[√2] is Principal Ideal Domain. True.

Q: 2. Determine whether or not the set is an ideal in the ring M2×2(R). J-{(8₂9) | PCR} = ER

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Q: I is an ideal of Z[x].

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Q: Let R = Z[x] and let P = {f element of R | f(0) is an even integer}. Show that P is a prime ideal of…

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Q: 21. Show that the set of polynomials {C₁, C₂ (2x - 1), C3 (6x² − 6x + 1)} in P(R) is orthogonal with…

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