Consider the irreducible polynomial p(x) = x² + x + 2 e Zs[x] and consider the simple extension field Zs[a], where a is a zero of the polynomial p(x). Then (a + 2)-! = 3a +4 2a + 2 O a + 4 O None O 4a + 2 O

Q: Consider the polynomial p(z) = z 101 + 7· z² + 3 · i · z – 1. Using Rouche's Theorem, determine how…

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Q: If q(x) is an arbitrary polynomial in Pn it follows that                                      q(x) =…

A: Given, if q(x) is an arbitrary polynomial in Pn it follows that                   q(x) = c0p0(x) + .…

Q: For an irreducible polynomial p(X) = 1+X + X² € F[X] consider F4 generated by p(X). Find x1, x2 E F,…

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Q: Let R[x] be the set of all polynomials with real coefficients. In other words, x] = {anx" + an-1x"-1…

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Q: Let f(x) = Q[x] be an irreducible polynomial and assume that f(x) splits in R. Let y E R be a root…

A: To prove that 2^(2/3) + gamma is a primitive element of Q(gamma, 2^(2/3))/Q, we need to show that it…

Q: 1. Find irr(a, Q) and deg(a, Q) for the given algebraic number a = √2+ i C. [Hint: Find irr(a, Q) as…

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Q: Let α be a zero of f (x) = x2 + 2x + 2 in some extension field of Z3.Find the other zero of f (x) in…

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Q: Determine the irreducible polynomial for 15 over each of the following (c) a = /3 fields (i). Q (ii)…

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Q: Let p: R → R be a polynomial of odd degrees. Prove that the equation p(x) = 0 has a solution in real…

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Q: 2. Let char F = 0 and f(x) E F[x] be a monic polynomial having positive degree. If d(x) = gcd(f(x),…

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Q: Now suppose b +0. The characteristic polynomial of A is (a-X)(d-A)-b² = x² − (a+d)λ + ad-b². From…

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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: Consider the homogenous polynomial ai + 10r1r2 + x². Find a suitable substitution s = Question 3 azi…

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Q: Let α be a zero of f (x) = x2 + 2x + 2 in some extension field of Z3.Find the other zero of f (x) in…

A: Given: α is a zero of fx=x2+2x+2 in some extension field of Z3. We have to find the other zero of…

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Q: Let fx)=x+x +2 be a Polynomial in Z3[x]. Which of the following statements is True O f(x) is…

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Q: The Taylor polynomial of degree 3 centered at x=0 of the function f defined by f(x)=sen(8x)+cos(9x?)…

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Q: 2. Consider the Newton polynomial Pn(x) = ao+a1(xr-xo) + a₂(x-xo)(x − x₁) +... +an(x-xo)(x -…

A: Given that   that interpolate then we have Now  ,       putting

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A: We will use GRAM-SCHMIDT ORTHOGONALIZATION PROCESS to solve this problem For a basis y0,y1,y2 to…