(a) Assume p is prime. Show that there are (p- 1)/2 irreducible polynomials of the form f(x) = x² – b in Z,[r]. (b) Show that for every prime p, there exists a field with p2 elements.

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

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Q: Suppose that a and b belong to a field of order 8 and that a2 + ab + b2 = 0. Prove that a = 0 and b…

A: Given: Suppose that a and b belong to a field of order 8 and that a2 + ab + b2 = 0. To Prove: a=0 ,…

Q: Prove that F2 = {0,1} is a field

A: Given: F2=0,1. We have to prove that F2=0,1 is a field.

Q: Let K be a field of characteristic zero. Show that a polynomial of the form t4+bt² + c is solvable…

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Q: Let F be a finite field with q elements and -1 -1] A 1 0 E € GL₂(F). Find N(A).

A: Definition: The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column…

Q: Let F be a finite field of order q and let a be a nonzero element in F.If n divides q - 1, prove…

A: In this problem, we are given a finite field F of order q, and a nonzero element a in F. We are…

Q: Let F = {0, 1,2} denote the field of 3 elements. Consider the field K obtained by adjoining the…

A: Given the field of 3 elements F=0,1,2., F=m=3 So, characteristics of field F is 3.

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: here exists a decomposition field for every polynomial  of positive degree over a field.

A: We have to show that there exists a decomposition field for every polynomial of positive degree over…

Q: Let F be a finite field of order q and let a be a nonzero element in F.If n divides q - 1, prove…

A: It is given that, F be a finite field of order q and let a be a nonzero element in F. We have to…

Q: Consider the polynomial x +x + 1 in Z2[x]. Which of the following statements is true? O This…

A: In this question , we have to use concept of Irreducible polynomials. Irreducible polynomials If a…

Q: Let F be a field of characteristic 2 with more than two elements.Show that (x + y)3 ≠ x3 + y3 for…

A: Let F be a field of characteristic 2 with more than two elements, and let x, y be arbitrary elements…

Q: et FCK be an algebraic field extension and let a K. Prove that if dimp(F(a)) is odd then F(a) F(a²).

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Q: For any prime p, show that there are no homomorphisms between F 2 and F 3 field P p in either…

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

A: As per the question we are given a nth order general polynomial f(x) over the field F[x] And we have…

Q: prove . If f (x) is any polynomial of degree n 21 over a field F, then there exists an extension K…

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Q: (9.1) Show that 3 ∈ Z15 is a zero divisor. (9.2) Prove that in any ring R, if a ∈ R is a zero…

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Q: Let F be a field of characteristic p ≠ 0. Show that the polynomialf (x) = xpn - x over F has…

A: Given that, F is a field of characteristic p≠0. It has to be proved that, over F, f(x)=xpn-x has…

Q: Let F be a field of characteristic p ≠ 0. Show that the polynomialf (x) = xpn - x over F has…

A: The given polynomial is fx=xpn-x. To Show: The above polynomial has distinct roots.

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: Find the splitting field E over Q, and a basis of E over Q of: (a) x*- 4 (6) x - 5

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Q: Let Q be the field of rational numbers and let F √2, √7, √14 are irrational. = Q(√2, √7). You may…

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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…

Q: Let F be a field and a(x), b(x) = F[x] not both zero. The greatest common divisor (gcd) of a(x) and…

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Q: Find all values of p such that Z„[x]/(x² + 1) is a field.

A: Given problem is :

Q: Find all values of a in Z7 such that the quotient ring Z7[x]/(p(x)) where p(x) = x³ + x² + ax + 3 is…

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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: Let F F, 2/2Z. Find an irreducible polynomial of degree 4 in Fla] and use it to construct a field…

A: In this question, it is given that  F = F2 = ℤ2ℤ . We have to find the irreducible polynomial of…

Q: Find an a such that (√2, √2) = Q(a). Show that your a is indeed in the given field. Verify by direct…

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Q: Let m and n be relatively prime positive integers. Prove that thesplitting field of xmn - 1 over Q…

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Q: (8) If F is a field, then it has no proper ideal. От F

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Q: Corollary - Let F be a field. Then F[x] is a unique factorization domain.

A: Given: Theorem 18.3: Every principal ideal domain is a unique factorization domain. Corollary: Let F…

Q: Consider the irreducible polynomial p(x) = x² + x + 2 e Zs[x] and consider the simple extension…

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