2.1. Suppose that F is a finite field with say |F| = pm that V has finite dimension n over F. Then find the order of GL(V). 9 and =

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: be a field and let c,d e F. Show that c · (-d) = -(c d).

A: Associative Property of Field F for a,b,c∈F a·b·c=a·b·c

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: 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: 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: /6. Let (R, +,) be an integral domain and consider the set Z1 of all integral multiples of the…

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Q: 5.10. Let K/F be a finite extension of fields. Prove that there is a finite set of elements a₁,...,…

A: A field K is said to be an extension of F if K contains F. The degree of K over F is the dimension…

Q: =Fnxn. Let C be a fixed element of by Tc(A) = AC-CA.

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

A: Solution: Given that

Q: 1. Prove the following property in an order field F. If r + y 0, then r 0 and y= 0.

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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: Find the divergence of the field F. F= -7x⁶i-5xyj-5xzk

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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: Show that the 8th cyclotomic field is Q(i, √2).

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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: Find the divergence of the field. F= 5z e 8xyz i+ 5x e 8xyzj + 5x e 8xyZk div F=O

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Q: Q: (A) Let R =(Z12, +12-12). 1. Find all maximal ideals of R. 2. Evaluate char. (R). 3. Is R a…

A: Introduction: Rings are algebraic structures in mathematics that generalise fields: multiplication…

Q: 5.9. (a) Let K/F be an extension of fields. Prove that [K: F] = 1 if and only if K= F. (b) Let L/F…

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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: 2.21 Find all possible Jordan forms of a transformation with characteristic polynomial (x-1)³(x+2).

A: The given characteristic polynomial x-13x+2. We have to find all the possible Jordan forms of the…

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: 14. Let F be a field of 8 elements and that a, b E F satisfy a² + ab + b² = 0. Prove that a = 0 and…

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Q: Let F be a field and let a, b e F. Show that (-a) - b= -(a - b).

A: Introduction: Associative property of field F for a,b,c∈F. (a·b)·c=a·(b·c)