9.22. Let F be a field, let f(x) = F[x] be a separable polynomial of degree n ≥ 1, and let K/F be a splitting field for f(x) over F. Prove the following implications: #G(K/F) = n! G(K/F) ≈ Sn f(x) is irreducible in F[x]. Note that the first implication is an “if and only if," but the second only goes in one direction. Show that the second implication cannot be reversed, by writing down an example for which it is false.

Q: Show that f(x) = x6 − 2ax³ + a is inseparable over Z3 (a), with a in any extension field of Z₂l

A: Given that                       f(x)=x6-2ax3+a We have  to show that f(x) is inseparable over Z3(a)…

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: 4) Let F be a field. The derivation f' (T) € F[T] of a polynomial f(T) E a; · Ti e F[T] is defined…

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Q: 3. Let L be a splitting field of a polynomial f(X) € F[X] of degree n. Show that deg L/F ≤n!.

A: Given that L is a splitting field of a polynomial f(x) ∈F[x] of degree n. We have to show that…

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

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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: Let α =21/8 and let K = Q(α,i). Show that (a) K is the splitting field of x8 −2 over Q and dimQ(K) =…

A: Given: α=218 and K=Qα,i.   To show: K is a splitting field of x8-2 over ℚ and dimℚK=16

Q: Let F be a field of characteristic Show that f(x) is irreducible over F or f(x) splits in F. p and…

A: Theorem Let p be a prime and F be a field of any characteristic.If the polynomial x^p-a\in F[x] has…

Q: Let f (x)= ax2 + bx + c ∈ Q[x]. Find a primitive element for thesplitting field for f (x) over Q.

A: Consider the given polynomial, fx=ax2+bx+c with a,b,c∈Q Let E be the splitting field of fx. if a=0,…

Q: i. Let F be a field. Prove that a polynomial p(X) = F[X] of degree ≤ 3 is irreducible if and only if…

A: Introduction: Let F be a field. We say that a non-constant polynomial f(x) is reducible over F or…

Q: Define an algebraically closed field. Show that field E is algebraically closed if and only if every…

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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: Prove that 2/27/a is a field where p(X) = X³ + X + 1 € Z/2z[X] and a = = (p(X)).

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Q: .55. (+) Let F be a field consisting of exactly three elements 0, 1, x. Prove that +x= 1 and that x…

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Q: (7) Let F be a field. Prove that (−1) · x = -X.

A: Let F be a field. We have to prove that (-1)·x=-x.

Q: Let K be an extension of a field F and f(x) be a polynomial of positive degree over F, then alpha∈K…

A: be an extension of a field and is a polynomial of positive degree over .We have to prove that  is a…

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

A: Given:  f(x) = x2 +2x + 2

Q: (6) Let F be a field and x € F. Prove that -(-x) = = X.

A: Solution: Given that

Q: 3. Let F be a field of characteristic #2. State and prove a necessary and sufficient condition on a,…

A: Sol:-A necessary and sufficient condition on α and β in F so that Fα = Fβ is that the elements α and…

Q: نقطة واحدة If f(z) = evsin 2z. Then 4 d Vsin 2z Vcot 2z. evain 2z %3D dz TO FO

A: ANSWER 2 :-

Q: 5.Let F be a field of char(F)=2. Then the number of elements xe F such that x = x is infinite

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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: 3. Suppose F is a splitting field of æ" – 1 over Z3. (a) Find |F| if n = 3. (b) Find |F| ifn = 13.…

A: Given polynomial is  Let p(x)=x^n - 1

Q: 8. Let F be a field and let f(x) be an irreducible polynomial in F[x] Then if the characteristic of…

A: Let f(x) be an irreducible polynomial in Fx. The characteristic of F is zero. To show: f(x) has no…

Q: Let I/Q be an extension of fields and f(X) = K[X] is an irreducible polynomial such that the degree…

A: Given:  LQ as an extension of fields and fX∈KX is an irreducible polynomial such that the degree of…

Q: .3. Let K be an extension of a field F. Let

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Q: Let F be a field and let f(x), g(x), n(x). k(x), and h(x) be polynomials in F[x]. Prove that if f(x)…

A: Given thatLet F be a field and let f(x) , g(x) , n(x) , k(x) , and h(x) be polynomials in F[x] .We…

Q: 3. Let F be a field and let a(x), b(x), c(x), d(x), and e(x) be polynomials in F[x] with a(x) ‡ 0F.…

A: To show that e(x) divides a(x)d(x), we need to prove that there exists a polynomial q(x) such that…

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

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Q: Let F = {0, 1, x, y, z} be a field with 5 elements. Given that x y = 1, find x . z.

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Q: Show that Q(√3, i) is the splitting field for x4 - 2x² - 3 over Q.

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Q: Find the divergence of the field F. F = yj-xk (y2, x2 1/2 x2- y2 (y2+ x23/2 -x2 ( 2+ x33/2 x2…

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Q: Let f (x) = ax2 + bx + c ∈ Q[x]. Find a primitive element for thesplitting field for f (x) over Q.

A: Given Data The function is f(x)=ax²+bx+c∈ Q [x]   Let a=0, The function is,