Let n > 1 be an integer. We say that [x] ∈ℤ/n is a zero divisor if there exists [y] ∈ ℤ/n − {[0]} such that [x][y] = [0]. We say that [x] is a unit if there exists [y] ∈ ℤ/n such that [x][y] = [1]. 7a) What are all the units in ℤ/12? What are all the zero divisors? b) What are all the units in ℤ/14? What are all the zero divisors? c) What are all the units in ℤ/13? What are all the zero divisors?

Q: Let G= Z* and let * be an operation in a * b. Is a * b = ab/2 is closed under the operation?

A: Disproof by an example.

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A: Consider the given information:Let a,b,x,y∈ℤ be such that ax+ by=11.

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

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Q: 3. A = {2, 2+2, 2+4,2+6, 2+8} where A = {2,4,6,8} 4. A=(1-22, 1+22) where A = Z*

A: Note : (i) ∩α∈S Aα=x : ∀α∈S ,x ∈Aα               (ii) ∪α∈S Aα=x : ∃α∈S ,x ∈Aα

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Q: (2) Suppose that G, C Zn-r. Then prove that G,-1C Zn-r+1

A: We prove that

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A: Given, x3≡amod p At p=7 We know that suppose p is rational prime p≡1mod 3 then x3≡a mod p is…

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Q: Prove that, IXI<a Rf and If -aL X Z a

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Q: For each subfield S of Q[i, z], determine Auts(Q[i, z]).

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Q: Q. Let X = {m, n}, and = {ø, X, {m},{n}}. %3D Define the function un: -[0, 00) by 3 if n e A 0 if n¢…

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Q: 17. Use Theorem 4.7 to prove that Vx + 3 2 V3 + x/4 for

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Q: Prove that Z Q has the universal property of inverting non-zero elements.

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Q: Let a and b be integers which are not both zero, and let d= gcd(a, b). Prove that gcd (a/d,b/d)= 1.

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