6) There is a multiplicative inverse for (2x+3) in Z₁[x] because (ax+3b) (2x+3)=1 where A = and b = Thaus in 71
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pleaseeeeeee solve question 6
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- Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.Use Theorem to show that each of the following polynomials is irreducible over the field of rational numbers. Theorem Irreducibility of in Suppose is a polynomial of positive degree with integral coefficients and is a prime integer that does not divide. Let Where for If is irreducible in then is irreducible in .Show that x4 + x + 1 over Z2 does not have any multiple zeros inany extension field of Z2.
- The Galios group of x^n-1 over any field of characterstics zero is abelian.Find the splitting field ofx4 + x2 + 1 = (x2 + x + 1)(x2 - x + 1)over Q.show that Q[x]/(3x⁴+2x³+1) is a field. Here (3x⁴+2x³+1) is the principal ideal generated by a polynomial 3x⁴ + 2x³ + 1 ∈ Q[x]. [Hint: Eisenstein Criterion.]
- Suppose that a and b belong to a field of order 8 and that a2 + ab + b2 = 0. Prove that a = 0 and b = 0. Do the same when the field hasorder 2n with n odd.Show that f(x)=x\power{6}-2ax\power{3}+a is inseparable over Z\index{3} (a), with a in any extension field of Z\index{3}.Let F be a field of characteristic 2 with more than two elements.Show that (x + y)3 ≠ x3 + y3 for some x and y in F.