Show that the Galois group of a polynomial of degree n has orderdividing n!

Q: The splitting field of f=(x^2-2)(x^2+1) over ℚ is ℚ(sqrt(2),i) because its roots are…

A: Please find the detailed solution in next step.

Q: Find a basis for span(l - x, x - x2, 1 - x2, 1 - 2x + x2) in P 2

A: We know that Pn is the vector space of all polynomials of degree less than or equal to n. This…

Q: What is the Galois group of x^4-5x^2+6 over ℚ?   ℤ_4 V_4 D_4 S_4

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Q: If n be any given positive integer, show that the mapping f:C\index{0}\rightarrow{}C\index{0}…

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Q: 9. Describe the group of the polynomial (x* – 1) e Q[x] over Q.

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Q: 12. Describe the group of the polynomial (x* – 5x? + 6) € Q[x] o over Q.

A: please see the next step for solution

Q: Q5:(A) Let R=(Z12 ,+12, -12 ). Find a) Maximal ideals of R b) Is (Z12, +12) a simple group? c) Is…

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Q: Show that the Galois group of x" - 1 over any field of characteristic zero is abelian.

A: We have to show that the Galois group of xn -1 over any field of characteristic zero is abelian.

Q: [Zp-(0),.] Where p is prime is an abelian group

A: We have to show that [Zp-(0),.] Where p is prime is an abelian group

Q: 1. What are the possible automorphism groups of the splitting field of an irreducible polynomial of…

A: Introduction: Galois group is one of the very significant group in extended field theory. In every…

Q: et K be the splitting field of (x^n-a)∈F[x]. Then  G(K,F) is a solvable group

A: If F contains a primitive nth root of unity, then G(K,F) is abelian and hence it is solvable. So,…

Q: Show that SL(2,R), the multiplicative group of 2x2 matrices over the real numbers of determinant 1,…

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: 9. Let T be an operator on an n-dimensional vector space V over a field F. (a) If T has n distinct…

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Q: 6. a) Prove that S, is not abelian for all n23. b) Prove that A, is not abelian for all n24. Ay…

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Q: Let G = .  Show that the permutations (23) and (13) satisfy the defining relations of G.  Explain…

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Q: The endomorphism A:Rn[x]->Rn[x] acts according to the rule A-> p - p(0). Then the rank of the…

A: Step 1:na Step 2: Step 3: Step 4:

Q: We cansider the field E= Q(V2, 3,13). It is shown that [E:Q]-8. we have the following cun jugation…

A: Given field is E=ℚ2,3,5 It is shown that E:ℚ=8 Conjugation isomorphism are…

Q: There exists a quartic polynomial (i.e. a polynomial of degree 4) over Q whose Galois group is…

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Q: What is the kernel of homomorphism of \phi:Z4->Z10 defined by \phi(x)=5x?

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Q: The center of GL(n, F). Let F be a field (the rationals, the reals, the complexes, or Z/pZ). Show…

A: Given GLn,F be the general linear group of all invertible matrices whose entries belongs to the…

Q: all possible Jordan Canonical Form of A if the characteristic poly- al is given by (1 – x)*(2 – x)².

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Q: Question 3: Leb 3:6>G' be an isomorphism . Prove the follawing assertions : la) IGl =1G| (b) lal =I…

A: We have to solve given problem:

Q: b) Let M = 0(2) (O(n) is the orthogonal group) Calculate the tangent space TM and the normal space…

A: The set of matrices is a manifold. The tangent space to a manifold at a point a is the null space of…

Q: Obtain the splitting field of

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Q: If N and K are submodules of M. (i) The kernel of the natural homomorphism N--> M/K is (N+K)/K. (ii)…

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Q: K/Q

A: According to the question,  a) We need to prove that the field extension K/Q is a Galois…

Q: The Galios group of x^n-1 over any field of characterstics zero is abelian

A: Let K be the splitting field of xn-1 over field F of characterstic zero.Now let  P be a primitive…

Q: 1. There is no simple group of order 200.

A: Solution:-As per guidelines I submit first question only

Q: Check whether or not if L/K and K/F are Galois extensions, then L/F is a Galois extension.

A: Suppose that EF and KF are extensions of a common base field F. Say that a field homomorphism φ: E→K…