Let K be the splitting field of f(x) = F(x)and let p(x) be an irreducible factor off(x) in F(x). If the roots of p(x) area₁, A2,... 0₁, then for each i there existsan automorphism o¡ in G(K, F) such that0₁(α₁) = 0₁.i

Since p(x) is an irreducible factor of f(x)∈F[x], then every root of p(x) is a root of f(x), so it…

Answered: Let 'R be the splitting field of f(x) €…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Prove Theorem 4.
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Theorem 7: Let F/k be an extension field and ae F be algebraic over k. Then the following holds: i) k[x]/(m₂(x)) = k[a]; ii) k[a] =k(α); iii) [k[a]:k] = deg α. Proof: Let :k[x]→F:o(f(x))=f(a), be the evaluation map at a. You should check that is a well-defined ring homomorphism. Let deg α = n. i) Since a is algebraic over k, Kero is a non-trivial ideal of k[x]. We claim that Ker = (m₁(x)). Since m(a) = 0, and Ker o consists of all the non-zero polynomials in k[x] satisfied by a, we see that m(x) = Ker . Hence (m₁ (x)) ≤ Ker o. C Request explain where has Now, let g(x) = Ker , then g(a) = 0 and so p(x)|g(x) in k[x]. Consequently, g(x) = (m(x)), and hence Kero ≤(m(x)), proving the claim. Now, by the Fundamental Theorem of Ring Homomorphisms, k[x]/ K[X]{m, (x)) ≈ Im ✨ = k[c]. o ii) By (1) above, k[x]/(m²(x)) = k[c]. Since m. (x) is reducible, k[x)/(m₂(x)) coincide with its quotient field, i.e., k[a] = k(α). is a field. Consequently, k[a] is a field. Therefore, it must pers come from
(b) Let A be an operator on a Hilbert Show that A is normal if and only if ||Ax|| = |A*x|| for every xe H.' space Н.
Construct an isomorphism between K and L, where K is the splitting field of x3 - x - 1 andL is the splitting field of x3 – x+1 | over F3.
Let f(x) EZ[x] be an irreducible polynomial of degree 4 such that it isGalois group over Q is isomorphic to S-4. Let a be a root of f(x). Show that Q(a) has no sub fields other than Q and Q(a).
If E is a splitting field of a polynomial f(x)∈F [x] any isomorphism between subfields of E over F can be extended to an F-automorphism.
If E is an extension of F and f(x) belong to F(x] and if phi is an automorphism of E leaving every element of F fixed prove that phi must take a root of f(x) in E into a root of f(x) in E
Define a ring homomorphism 4: R[x] → RX R, (ƒ(x)) = (fƒ(0), ƒ(1)) In other words, y maps f(x) to its evaluations at x = and at x = = 1. (a) Describe the kernel of y, in terms of polynomials having certain roots/zeroes. (b) Find polynomials f(x) and g(x) in R[x] such that f(0) = 1, f(1) = 0, and g(0) = 0, g(1) = 1 (Note: There are many different options!) Let a, b € R, and define h(x) = af (x) + bg(x). Show that y(h(x)) = (a,b). (c) Using part (b), explain why is surjective. (d) What does the 1st Isomorphism Theorem say, when applied to ?
If p(x)∈F[x] and deg p(x) = n, show that the splitting field for p(x)over F has degree at most n!.

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Let 'R be the splitting field of f(x) € F and let p(x) be an irreducible factor of f(x) in F(x). If the roots of p(x) are α₁, α2,... O¡, then for each i there exists an automorphism o; in G(K, F) such th