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

Q: What is the relationship between the degree of a field extension and the size of its Galois group?

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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: The group (Z4 ⨁ Z12)/<(2, 2)> is isomorphic to one of Z8, Z4 ⨁ Z2, orZ2 ⨁ Z2 ⨁ Z2. Determine…

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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.

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Q: group ⟨a,b,c|b^4 = a,c^(−1) = b⟩ is abelian.

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Q: (a). H, K≤ G and HGH* K≤G. (b). HG and KAG⇒ H✩ KAG.

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Q: Show that x4 + x + 1 over Z2 does not have any multiple zeros inany extension field of Z2.

A: We have given that, fx=x4+x+1  We need to show that fx does not have any multiple zeroes in any…

Q: Give the splitting field of x4 - 1 over 1 over Z2, Z3 and Z5.

A: Given: Objective: To give the splitting field of  over  and .

Q: Show that f(x)=x\power{6}-2ax\power{3}+a  is inseparable  over Z\index{3} (a), with a in any…

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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 (G1, -) and (G2 , *) be two groups and o : G1→ G2 be an isomorphism. Then * G2 might be abelian…

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Q: polynomial in Z_2[x] (where Z_2 is the field containing two elements 0 and 1)? a) x^2 b) x^2+x c)…

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Q: The Galois group of x\power{n}-1 over any field of   characteristic zero is abelian.

A: Let K be the splitting field of xn-1 over a field F of characteristic zero. Let ξ be a primitive…

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Q: prove: let g be a group, if g is abelian then (ab)^2 = (a^2)(b^2)

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Q: True or false, The nonzero elements of ℚ form a cyclic group ℚ* under field multiplication.

A: Cyclic group under Multiplication

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A: Suppose K is a cyclic extension of degree n over F.Let r ∈F be primitive nth root of unity , and σ…

Q: Suppose that F is a field of order 125 and F* = <a>. Show thatα62 = -1.

A: It is given that F is a field of order 125 and F*=α. It is required to show that α62=-1. Let F*=α…

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Q: 3. Let D be an integral domain. (a) Prove that FD is an abelian group under the operation of…

A: Let us prove that field of fractions is an abeliean group under addition.

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Q: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

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Q: 2. Finite, abelian (and additive), not isomorphic to Z₂ or Z2 x Z2, and also for every element a,…

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