4. Let w = e 11, and let E = Q(w).(a) Verify that [E:Q]10.(b) Determine |Gal(E/Q)|, the number of automorphisms of E thatfix Q.(c) Find the order of y(w)w? in Gal(E/Q).(d) How many subfields K does E have (such that QC K C E)?Hint 1 The corollary to Eisenstein's theorem might help.Hint 2: Example 6, page 535 may help.

Answered: 4. Let w = e, and let E = Q(w). (a)…

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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Express the following using language of Predicate Calculus, where it is understood that the people being discussed are in the courtroom If any sentence is ambiguous, give all symbolic versions. (i) All judges are sober (ii) There is a dishonest lawyer. (iii) All defendants are innocent. (iv) Some plaintiffs are lawyers (v) Anybody who is honest and a defendant is innocent (vi) All defendants who are not sober are dishonest.
1. Show that which of these relations on the set of all functions on Z→Z are equivalence relations? (a) R = {(S,8)|S (1) –g(1)} (b) R = {(f,g)|f (0) = g (0) or f(1)=g(1)}
1 Which of the following complex functions is holomorphic in C? (A) f(x + iy) = y for every x E R, y E R (B) f(x+iy) = x² – y? + i2xy for every x E R, y ER (C) f(x+iy) = y + ix for every x E R, y E R (D) f(x+iy) = x² + y? + i2 for every x E R, y E R
(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.
6. Determine whether each of these functions is a bijection from R to R. a) f (x) = x² + 1 b) f (x) = x³
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5. Let W be the space of polynomials p(x) with real coefficients of degree < 10, satisfying p(0) = p(1) = p'(1) = 0. The dimension of W is (a) 1 (b) 7 (c) 8 (d) 9 (е) 11
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3. Consider the set l' = {x = (x1, x2, x3,...) E C : i=1 consisting of all absolutely summable infinite sequences of complex numbers. a) Show that l1 is a complex vector space. b) Show that 8. ||x||1 = la| 2=1 defines a norm on l.
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Why is inversion [that is, Φ(x, y) = (-x, -y)] not listed as one ofthe four kinds of isometries in R2?

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