15. (a) Show that S= {(a,a) | a e Z} is a subring of ZxZ. (Use the definition of addition and multiplication of direct products.) (b) Determine if T = {(a,-a) | ae Z} is a subring of ZxZ. 16 Let B he a commutative ring with unity and let

Q: (15 points) Prove the ring Z[√−2] = {a+b√−2 | a,b ≤ Z} is a Euclidean domain with norm 8(a+b√2) = a²…

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Q: 9. Let R be the ring Q[x]/(x³ - x² − x + 1). (a) List all the ideals of R and draw a diagram showing…

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Q: 4. Find a monic polynomial of least degree in ℝ[x] that has 1-i and 2i as roots. write the powers of…

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Q: 7. Let R be a commutative ring and let F = F(R, R) be the set of functions f: R→ R. Functions in F…

A: let R be a commutative ring and F=FR,R be the set of function f:R→R defined as f+gx=fx+gxfgx=fxgx…

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A: Ans- (a) To show that S = {n1R ∈ R : n ∈ Z} is a subring of R, we need to show that S is non-empty,…

Q: 5. Explain why each of the polynomials below is irreducible in the indicated polynomial ring. Make…

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Q: Let R be a ring. (a) Show that R has exactly one zero, 0R. (b) If R also has an identity, 1R, show…

A: Given: R is a ring.We need to prove the following:(a) Show that R has exactly one zero, 0R. (b) If R…

Q: 10. Decompose x* + 4.x² + 1 into a product of irreducible polynomials in the following rings: (a)…

A: To find - Decompose x4 + 4x2 + 1 into a product of irreducible polynomials in the following rings :…

Q: Enter the multiplicative inverse of the element [21]49 X+ [18]49 in the ring Z49[x].

A: Extended Euclidean Algorithm

Q: 4. Consider P = P(3, V15) = {a + b/15|a, be Z, 3|a} and Z[V15] = {a + bv15|a, b e Z}. (a) Prove that…

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Q: B E None (i), (ii) (iii), (iv) Which of the following are ring homomorphisms? (i): f: (Q(√2), +,-) →…

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Q: (a) i. Explain why the ring Z4 is not a field. ii. Explain why the ring Z₂[x]/(x² + 1) is not a…

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