Let f(x)=x^4-4. a) Determine the degree of the splitting field K of f(x) over Q. b) Determine all the sub extensions F such that Q is contained in F and F is contained in K, and arrange these in a graph ordered by inclusion.

Q: Let f (x) ∈ F[x] and let a ∈ F. Show that f (x) and f (x + a) have thesame splitting field over F.

A: Given that fx∈Fx and a∈F We have to show that fx and fx+a have the same splitting field over F.

Q: Let F(a) be the field described in Exercise 8. Show that a2 and a2 + a are zeros of x3 + x + 1.

A: Given: Fa is a field. To show: a2 and a2+a are the zeros of x3+x+1.

Q: 11. Find the greatest common divisor of  x5 + x4 + x3 + x2 + x + 1 and x3 + x2 + x + 1 in F[x],…

A: We find the greatest common divisor of  x5 + x4 + x3 + x2 + x + 1 and x3 + x2 + x + 1 in F[x] i.e…

Q: Exhibit the finite field F_16 in the form F_2[x]/(p(x)) for some polynomial p. Identify all elements…

A: To exhibit the finite field F16 asF2xpx. Please follow the next step. '

Q: 2 Let m = R[x] be a polynomial with deg m≥ 1. Define a relation Sm on R[x] by the rule that (f,g) €…

A: An equivalence relation is a binary relation on a set that satisfies three properties: reflexivity,…

Q: Let F be a field and let f(x) = a,x" + a,-1x"-1 + · ·· + a, E F[x]. Prove that x - 1 is a factor of…

A:

Q: Show the map f : Q[x] → Q(√3) defined by f(p(x)) = p(√3) is a ring homomorphism. FInd its kernel.

A: Given that the map  defined as  .Let  then  and  where .Now, consider and hence,  is ring…

Q: here exists a decomposition field for every polynomial  of positive degree over a field.

A: We have to show that there exists a decomposition field for every polynomial of positive degree over…

Q: Let F be a field and let R be the integral domain in F[x] generated byx2 and x3. (That is, R is…

A:

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

A:

Q: Let F be a field which contains a primitive nth root of unity. If K is cyclic extension of degree n…

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

Q: Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily…

A: Given: f(x) is a polynomial of degree n over field F. let n=1 and f(x)=ax+b here a,b=0. Roots of…

Q: Let Q be the field of rational numbers and let F √2, √7, √14 are irrational. = Q(√2, √7). You may…

A:

Q: 31. Show that if F, E, and K are fields with F < E < K, then K is algebraic over F if and only if E…

A: We have to prove that using algebraic extension.

Q: Let f (x) be a cubic irreducible over Z2. Prove that the splitting fieldof f (x) over Z2 has order…

A: It is given that, f(x) be a cubic irreducible polynomial over ℤ2. We have to prove that the…

Q: Corollary - Let F be a field. Then F[x] is a unique factorization domain.

A: Given: Theorem 18.3: Every principal ideal domain is a unique factorization domain. Corollary: Let F…

Q: If F is a field containing an infinite number of distinct elements, the mapping f → f~ is an…

A: