Let K be any field extension of a field F and let an element a of K be algebraic over F.Then for any F – automorphismover F.o of K o(a) is a conjugate of a

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Answered: Let K be any field extension of a field…

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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Let F/k be an extension field of finite degree. Show that every k-endomorphism of F is an automorphism of F. Give an example, with justification to show that this may fail to hold for arbitrary extensions.
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a. Determine which one of the following statements is true (or false): Any unity is a unit. Any integral domain is a field. b. Let Z9 {[0] , [1] , [2] , ..., [8]} be the set of all equivalence classes of the relation =9 on Z, which is defined by a =9 b + 9 | (a – b) for any a, b e Z so that the equivalence class of k E Z is [k] = {x € Z | x =9 k}. Recall that Zg together with the binary operations + and ·, which are given by [a] + [6] = [a + b] [a] · [6] = [a · b] and for any [a, b E Zg, is a commutative ring with unity. i. The element is the additive identity. ii. The element is the unity. iii. The element is a unit. iv. The element is a zero divisor. V. The ring Zg is ? vi. The element 6 is ? vii. The element is a root of the polynomial f (X) = [5] · X² + [7].
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Let K be a field extension of a field F Then an element a € K is algebric over F if and only if F(a) is finite extension of F that is [F(a): F] is finite.
Let K be a finite normal extension of a field F. If a, ß are any two elements of K conjugate over F, then their exists an F – automorphism & of K such that o(a) = ß -
Let K and E be the extension of a field F with [K:F] finite and assume that both K and E are subfields of some larger field. If K is a Galois extension of F then KE is a Glois extension of E and G(KE, E)≅G(K,K∩E).
Let E be the splitting field of x4 + 1 over Q. Find Gal(E/Q). Find allsubfields of E. Find the automorphisms of E that have fixed fieldsQ(√2), Q(√-2 ), and Q(i). Is there an automorphism of E whosefixed field is Q?
Let K be an extension of a field F. If a, be K are algebraic over F, then a ±b, ab, ab' (b# 0) are algebraic over F. In other words, the elements of K which are algebraic over F form a subfield of K.

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Let K be any field extension of a field F and let an element a of K be algebraic over F. o of K o(a) is a conjugate of a Then for any F – automorphism over F