(i) Find gcd(2,3+5i) in Z[i], (ii) Show that 3 + 4i and 4-3i are associates in Z[i] (iii) Determine the splitting field of x4 + x2 + 1 over Q also find its degree over Q,

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A: Given A pair of complex numbers as followsAimTo write the complex numbers in simplified form and…

Q: (i) find gcd(2,3+5i) in Z[i], (ii) show that 3+4i and 4-3i are associates in Z[i] (iii) Determine…

A: (i).

Q: 2. Find all of the prime quadratic polynomials in F4[x], where F4 {0,1, a, b} be the field of four…

A: Given: F4={0,1,a,b}

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A:

Q: Consider the quotient ring R1 = Z3[a]/(x² + 1) and R2 = Z3[i] = {a + bi / a,b e Z3} with operations…

A: Consider the quotient ring R1=Z3x/x2+1  and  R2=Z3i=a+bi/a,b∈Z3 with operations…

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A:

Q: Find a splitting field S of the polynomial x^4-7x^2+10 over Q. Find [S:Q] and a basis for S/Q.

A: The given polynomial is x4-7x2+10 over the field of rational numbers ℚ.

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.

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A:

Q: Consider a basis C= {[3] [3]} Let (41c = [-2] What is in standard coordinates? of Rn

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Q: Prove that if the complex number z is a root of polynomial f(x) with real co-ffiecents , then z bar…

A: Solution

Q: (B) Find all maximal ideals of (Z8, +8,8).

A:

Q: (b) The characteristic polynomial of A is (A² + 4 +13)²(x - 5)². find all possible Jordan canonical…

A:

Q: 2c

A: The answer has been provided.Explanation:

Q: For the number field K = Q(V7): (a) Find the minimal polynomial of V7 over Q. (b) Find all three…

A:

Q: 4. (a) Solve the set of congruences х 1(mod 3), x= 2(mod 4), x = 3(mod 5).

A: We  will solve it by Chinese Remainder Theorem Given  x≡1(mod3)x≡2(mod4)x≡3(mod5) Compare it with…

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Q: Give the smallest extension of Q that contains all the zeros of (x4 – 8x² + 15)(x² + 1). -

A: Yes

Q: Show that (p(x), g(x)) = p(0)q(0) +p(1)q(1) + p(2)q(2) defines an inner product on P2.

A:

Q: 2d

A: Definition of P2​(R): Space of polynomials of degree at most 2 with real coefficients.Introduction…

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A: Given that α be a zero of x3+x2+1 in some extension field of ℤ2. We have to solve the equation…