Let R= Z/5Z, the integers mod 5. The ring of Gaussian integers mod 5 is defined by R[i] = {a+ bi : a, b e Z/5Z and i = v-1}. Show that R[i] is not an integral domain (and hence not a field) by showing that 3+i is a zero-divisor in R[i].

Q: 4. Show that 7 is irreducible in the ring Z[V5] using the norm N defined by N(a + bv5) = | a? –…

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Q: Suppose that a is algebraic over a field F. Show that a and 1 + a-1have the same degree over F.

A: Let's consider that 1+a-1∈Qa then, it can be concluded that, Q1+a-1⊆Qa                  …1

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A: Given: Objective: To give the splitting field of  over  and .

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Q: 11. Find the greatest common divisor of  x5 + x4 + x3 + x2 + x + 1 and x3 + x2 + x + 1 in F[x],…

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Q: Let Z,[i] = {a + bi |a, bƐZ,} (see Example 9 in Chapter 13). Show that the field Z,[i] is…

A: Given: Z3i=a+ib  a , b∈Z2   To Show: Field Z3i is ring -isomorphic to the field Z3xx2+1.

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Q: 5. Consider the ring Q[/2] (with the natural addition and multiplication). Is it an integral domain?…

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Q: Let R= Z/3Z, the integers mod 3. The ring of Gaussian integers mod 3 is defined by R[i] = {a+ bi :…

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Q: Let R = Z/3Z, the integers mod 3. The ring of Gaussian integers mod 3 is defined by R[i] = { a + bi…

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

A: Extended Euclidean Algorithm

Q: Appendix C: Fields Problem 1. Consider the following set of complex numbers Q(i) = (a + bi: a, b €…

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Q: 9. Let Zs is a ring of integers mod 5 and f(x)eZ,(x) such that f(x) = x²+x+1, then: Zs a) f(x) is a…

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Q: 30. Let B; be o-fields of subsets of 2 for i = 1, 2. Show that the o-field B1V B2 defined to be the…

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Q: Let R = ℤ/3ℤ, the integers mod 3. The ring of Gaussian integers mod 3 is defined by R[i] = { a + bi…

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Q: 39 In Z,[x], let I = (r + x + 2). Find the multiplicative inverse of 2x + 3+I inZ.[x\/I.,

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Q: Let Z3[i] = {a + bi : a, b ∈ Z3, i2 = −1} where arithmetic is done mod 3. 1.  Is Z3[i] an integral…

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Q: Write an equation for the function graphed below 15+ 14+ 13+ 12+ + + + -7 -6 -5 -4 -3 -2 -1 -1 -2 -3…

A: A following graph:To Find:Equation of the above graph.

Q: For which fields F listed below is the polynomial X + X3 +1 € F[X] irreducible? Select all that…

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