(a) Rings that are not integral domains (b) Integral domains that are not fields (c) Integral domains of characteristic 7
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A: Thanks for the question :)And your upvote will be really appreciable ;)
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A: Thanks for the question :)And your upvote will be really appreciable ;)
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Q: (3) If R is an integral domain has non zero characteristic, then Char(R)=..... (i) 5 (ii) 4 (iii)9
A: Option (i)
![(a) Rings that are not integral domains
(b) Integral domains that are not fields
(c) Integral domains of characteristic 7
(d) Prime ideals in a ring R that are not maximal ideals
(e) Units in Z15[x]
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- Prove the Unique Factorization Theorem in (Theorem). Theorem Unique Factorisation Theorem Every polynomial of positive degree over the field can be expressed as a product of its leading coefficient and a finite number of monic irreducible polynomials over . This factorization is unique except for the order of the factors.True or False Label each of the following statements as either true or false. 4. Any polynomial of positive degree over the field has exactly distinct zeros in .Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in
- True or False Label each of the following statements as either true or false. 8. Any polynomial of positive degree that is reducible over a field has at least one zero in .Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.a. Show that x3 + x2 + 1 is irreducible over ℤ3. b. Let ꭤ be a zero of x3 + x2 + 1 in an extension field of ℤ2. Show that x3 + x2 + 1 factors into three linear factors in (ℤ2(ꭤ))[x] by actually finding this factorization. [Hint: Every element of ℤ2(ꭤ) is of the form ꭤ0 + ꭤ1ꭤ + ꭤ2ꭤ2 for ꭤi = 0,1. Divide x3 + x2 + 1 by x - ꭤ by long division. Show that the quotient also has a zero in ℤ2(ꭤ) by simply trying the eight possible elements. Then complete the factorization)
- Factorization of polynomial over a UFD. If the integral domain R satisfies the ACCP, prove that the polynomial ring R[x] satisfies the ACCP.An irreducible polynomial f(x) over a field F of characteristic p>0 is inseparable if and only if f(x)=g(x\power{p}) i.e., f(x) is a polynomial in x\power{p}.1. Let R be a commutative ring with unity 1 and I is an ideal in R. Then what is the unity in the ring R/I?2. What is the degree of f=(20x+1)(5x+1) in ℤ_5[x]? Type only the numerical value if it is defined if not type undefined.
- a. Show that there exists an irreducible polynomial of degree 3 in ℤ3[x] . b. Show from part (a) that there exists a finite field of 27 elements.Prove that isomorphic integral domains have isomorphic fields of quotients. Definition of the field of quotients: F={a/b|a,b in R and b is not equal to 0}. Let R = Z. Find 5 examples of principal ideals. . Let R = R[x]. Find 5 examples of principal ideals . Consider the ring (P(Z), Δ,∩). What is ⟨{2}⟩?