In the ring Z2[x]: 1- The inverse of x is: a) x+1 b) x2 +1 c) x d) x2

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

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Q: If R= = b {(6) a. b. c real} and 1-{(5) b real). | a, I = (a) R is a ring. (b) I is an ideal of R.…

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Q: Explain why the polynomial rings R[r] and C[x] are not isomorphic.

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Q: Prove that Q[X]/(X2-2) and Q[X]/(X2-3) are non-isomorphic extensions of Q. In other words, you need…

A: We need to prove that  and  are non-isomorphic extensions of .We know that if is field and  is…

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

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Q: Exercises :- 1- Show that: (Z24 / {0,6, 12 ,18},+,.) = (Z6, +6 r6) 2- Show that: (Z20 /{0,10},+,.)…

A: "Since you have posted a question with multiple sub-parts, we'll solve first 3 parts for you. To get…

Q: 10. Determine if g(x) divides f(x) in the given rings of polynomials (show your work): a) g(x) x55x4…

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Q: In the ring Z[x]/ 1- The inverse of x is: a) x+1 b) x2 +1 c) x d) x? 2- The value that replaces x is…

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Q: a) x+1 b) x² +1 c) x d) x 2- The value that replacesx is a) x+1 b) x?+1 c) X d) x? 3- (x+1) is equal…

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Q: CE Z3 A) Find all the values that make the Z[x]/(x² +x + c) quotient ring an object. B) Show that…

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Q: In the ring Za[x]/ 1- The inverse of x is: a) x+1 b) x2+1 c) x d) x? 2- The value that replaces x…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: 1. Factor each of the following polynomials completely into irreducible polynomials: a) x² + 1 in…

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Q: Let a be a complex zero of x2 + x + 1 over Q. Prove thatQ(√ a)= Q(a).

A: Given equation is  x2+x+1 Let a be the complex roots over ℚ.

Q: In the ring Z[x] 1- The inverse of x is: a) x+1 b) x+1 c) x d) x 2- The value that replacesx is a)…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: The fields Q(√3) and Q(11) are isomorphic. O True O False

A: We have to determine that whether the following statement is true or false ; The fields Q33 and Q113…

Q: The number of reducible monic polynomials of degree 2 over Zz is

A: The number of reducible monic polynomials of degree 2 over Z3 is

Q: 1. Find the greatest common divisor of p(x) = x³ + 6x2 + 6x +5 and g(x) = 23x²-x-2. Does your answer…

A: To find the gcd of p(x) and q(x), we can use the Euclidean algorithm: First, p(x) /q(x) and  we get:…

Q: Let R be a ring with unity and assume p, q, r ∈ R∗ Find (pqr)-1and also prove that it’s the inverse…

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Q: Show that in the factor ring Z[i] / , the element x + is a unit.

A: Given a factor ring Zx2x+1=a0+a1x,+⋯+2x+1 |a1∈Z  As 2x+1+2x+1=0+2x+1 Since,  2x+1∈2x+1   as aH=H iff…

Q: -1 1 3 -5 1 4 7 Which of the following lines is fitted to the points using the least squares method?…

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Q: In the ring Za[x]/ 1- The inverse of x is: a) x+1 b) x2 +1 с) х d) x2 2- The value that replaces x'…

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Q: Show, by example, that for a commutative ring R, it is possible that deg(fg) < deg f + deg g where…

A: Given: fx and gx are non-zero elements in Rx. To prove: deg fxgx&lt;deg fx+deg gx by an example.

Q: Show that over the ring Z/30Z[x] the polynomial f(x)=x has the factorization…

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Q: In the ring Z[x]/ 1- The inverse of x² is a) x+1 b) x² +1 с) х d) x2 2- The value that replaces x is…

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Q: Simplify the following expressions in the indicated polynomial rings: A. (3x³ + 2x² + 4x + 2) + (2x¹…

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Q: If R is commutative ring then deg(F)g(x)) = deg(f(x)) + deg(g(x)). False O True

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Q: 1+xr 1+x* a) b)

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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.

Q: Use the field norm to show: a) 1 + √2 is a unit in Z[√2] b) −1+√√-3 -1+√√-3 2 is a unit in Z [ 1+…

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Q: We know that in Z3:      x4+ x3+ x + 1 = (x+1)4 How many distinct ideals does the ring   Z3[x] /…

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