Let f(x) = x² +1 € Z3[x] and let R= Z3[x]/I, where I = (f(x)). (a) Show that R is a field with 9 elements. (b) Denote by 0 := 0 + I, 1 := 1 +I, and a := x +I. Write the other 6 elements of R in terms of a and determine the multiplicative inverse of each nonzero element. (c) Prove that R=Z3[i].

Q: Find the dimension of the splitting field of each of polynomials x+1 and x² - 2.

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Q: Find the splitting field x4 - x2 - 2 over Z3.

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Q: any cubic polynomial f = K[r] can be written in the form f = r³+ar+ b. change of varillible

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Q: Write out the addition and multiplication tables for Z3 [x]/(x² + 1). Is it a field?

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Q: be a field and let c,d e F. Show that c · (-d) = -(c d).

A: Associative Property of Field F for a,b,c∈F a·b·c=a·b·c

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A: Since, α is a root of X2 - a = 0, we have          α2 - a = 0.  => α2 = a Since, β is a root of…

Q: Show by any means that Q[r]/(x4 + 3x2 + 9x + 1) is a field.

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Q: Let f(x)= x³ + 2x+ 1 and g(x)= 4x + 1 be two polynomials in a ring (Zs[x],+ ,.). Find q(x) and r(x).

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Q: Find the splitting field x4 - x2 - 2 over Z3.

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Q: Let A = 1 -X -X X 1 -x X X -1 Find x such that A is invertible.

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Q: Find the splitting field ofx4 + x2 + 1 = (x2 + x + 1)(x2 - x + 1)over Q.

A: The extension field K of a field F is called splitting field for the polynomial fx∈Fx if fx factors…

Q: Let F be a field and aeF be such that [F (a): F]=5. Show that F(a)= F(x³).

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Q: Let F be a field and let f(x) = a,x" + a,-1x"-1 + · ·· + a, E F[x]. Prove that x - 1 is a factor of…

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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: Let P(R#) represent the set of all polynomial functions, functions that can be written in the form…

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Q: Use the field F =Z2[x]/<x2 + x + 1> to construct a (5, 2) linearcode that will correct any…

A: Consider the given field, F=Z2x/x2+x+1 Now, abbreviate the coset a+x2+x+1 with a. Then the…

Q: Show that Z3[x]/<x2 +x + 1> is not a field.

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Q: (b) How many elements are there in S? Justify your answer.

A: The elements areNote: the elements of  are

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Q: (a) Show that Q(√3, i) is a simple exension of Q by verifying that Q(√3, i) = Q(√3+ i). (b) Show…

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Q: Consider the polynomials a(x) = r' + x³ + x + 4 and b(x) = 2r³ + 4.x² + x + 2 from Z5[r]. a) Find…

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Q: 2х1 + 3x2 — хҙ — 2 Зx1 — 2х2 + Xз — —1 -1 %3D -5x1 4x2 + 2x3 = 3

A: Consider the provided equations, 2x1+3x2-x3=23x1-2x2+x3=-1-5x1-4x2+2x3=3 Solve the above equation by…

Q: f = x² + x³x²+1, g = x³ + 1 in Q[x];

A: Since you have posted multiple questions, we will solve the first question for you. If you want any…

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: R = Z7[x] (x² +T) heck whether or not this is the splitting eld of a polynomial over Z₁.

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Q: Consider the field Z3[x]/. Please explain to me how this field has 9 elements. Thank you!

A: Consider the field Z3[x]/<x2 +x+2>. Please explain to me how this field has 9 elements.

Q: Find the splitting field ofx4 + x2 + 1 =  (x2 + x + 1)(x2 - x + 1)over Q.

A: Given: x4+x2+1=x2+x+1x2-x+1To find: The splitting field of x4+x2+1=x2+x+1x2-x+1 over Q.

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Q: Find the splitting field for f (x) = (x2 + x + 2)(x2 + 2x + 2) overZ3[x]. Write f (x) as a product…

A: Given: fx=x2+x+2x2+2x+2. We have to find the splitting field of f(x) over Z3x.Also, write f(x) as a…