7. Suppose R is an integral domain having the property that Rac] is a PID. Prove that R is a field as follows: (a) Explain why the substitution homomorphism o: R[x] →→R (that is, Ho (p(z)) = p(0)) is onto (b) Determine kerpo as a principal ideal (just write what it equals - you don't have to prove it). Why must kerjo be prime? (c) Now, what can we say about a non-zero prime ideal in a PID? (d) Make the conclusion that R is a field

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A: From the given information. The function f(x) is as follows.

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Q: 4. The following polynomial is irreducible in Q[x]: m(x) = x² - 2x²-1 € Q[x] Let's consider the…

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Q: What is the order of the splitting field ofx5 + x4 + 1 = (x2 + x + 1) . (x3 + x + 1) over Z2?

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Q: /6. Let (R, +,) be an integral domain and consider the set Z1 of all integral multiples of the…

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Q: Let F be a field of order 32. Show that the only subfields of F areF itself and {0, 1}.

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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 Z(V2) = {a + by2: a, b e Z}. Then Z(V2) is not a field not a commutative ring a field not an…

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Q: What is the order of the splitting field of x5 + x4 + 1 =  (x2 + x + 1) . (x3 + x + 1) over Z2 ?

A: The splitting field is : The aim is to find the order of the given splitting field over

Q: 3. Let F be a field of characteristic #2. State and prove a necessary and sufficient condition on a,…

A: Sol:-A necessary and sufficient condition on α and β in F so that Fα = Fβ is that the elements α and…

Q: 2. Let R[x] be a ring over field R and let f, g are elements of R[x]. f=x3 +x2 +x +[0] , g=x +[1].…

A: We are given : f(x)=x3+x2+x+0⇒f(x)=x3+x2+xand g(x)=x+[1]⇒g(x)=x+1Now, Dividing f(x) by  g(x), we…

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Q: What is the order of the splitting field of x³ + x² + 1 = (x² + x + 1) ⋅ (x³ + x + 1) over Z₂?

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Q: 9. Show that F= field? T a b a | a, b € Z5 is a subring of M2(Z5). Is Fa be Z5

A: Note : '' As per guidelines we will solve the first question. If you want any specific question to…

Q: Prove that Z[i]/ is a field.

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Q: Let F = {0, 1, x, y, z} be a field with 5 elements. Given that x y = 1, find x . z.

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Q: Consider the irreducible polynomial f = X' + X3 + X? +X +1 in 10. Z[X]. Let a = [X] in the ring…

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Q: Determine if R is (1) a field (2) an integral domain (3) a unital ring, where R= {2+y√P+2√z, y, ze…

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Q: Show that Q(√3, i) is the splitting field for x4 - 2x² - 3 over Q.

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Q: Find the divergence of the field F. F = yj-xk (y2, x2 1/2 x2- y2 (y2+ x23/2 -x2 ( 2+ x33/2 x2…

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Q: (b) Prove that Z/4Z is not a field.

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Q: The field is F,, the splitting field of x²-x, over F₂. Now x' - x = x(x - 1)(x² + x +...+x+1) = x(x…

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Q: Prove that 2/3Z Z, as rings,

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Q: Show that R[x]/<x2 +1> is a field.

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