3. If |F|q<∞ show:(a) There exists a prime p such that char F = p.(b) q = p some n.(c) a = a for all a € F.(d) If b € K is algebraic over F then ba = b for some m > 0.

3. a)We know that, Since F is a finite field, the characteristic of F must be a prime number p or…

Answered: 3. If |F|q<∞ show: (a) There exists 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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7. For each of the following relations on the set of all real numbers, solve all of the subparts as they are all connected rightly on a word processing document to determine whether it is reflexive, symmetric, antisymmetric, transitive. Here xRy if and only if: (a) x + 2y = 0 (b) x = 2y(c) x − y is a rational number (d) xy = 0(e) xy ≥ 0 (f) x = 1 or y = 1 (f) x = 1 or y = 1(g)x is a multiple of y (h) xy = 1
! 1 Express the interval shown on the number line in interval notation. F1 [] [₂) || G] Type here to search Interval Notation: F2 -12 -10 -8 -6 -4 -2 0 2 8 2 G] () A S F3 # m 3 E F4 yx $ 4 9 N F5 R F 4 % 5 6 W F6 -DO 8 T A 6 G 12 OT DELL F7 Y B F8 & 7 7 H 10 F9 U * A 8 8 N CO J
5. Let f: CC be the function defined by f(z) = 25 for any z = C. Fill in the blanks in the blocks below, all labelled by capital-letter Roman numerals, with appropriate words so that they give respectively a proof for the statement (E) and a proof for the statement (F). (The 'underline' for each blank bears. no definite relation with the length of the answer for that blank.) (a) Here we prove the statement (E): (E) The function f is surjective. [We want to verify the statement (II) For this C, (IV) By definition, z E C. Note that f(z) = 2 = It follows that (VI) (b) Here we prove the statement (F): (F) The function f is not injective. (III) [We want to verify the statement (II) Note that zo, wo E C. Also note that We have (IV) and It follows that (VI) (III) (V) (1) 9 such that CC(cos(0) + sin(0)). (1) (V) Then f(zo) f(wo).
75% For each of the following two statements, give a counterexample to show that the state- ment is false. (a) Let f: N + N be a function. If f is one-to-one, then f is onto. (b) If ACB and AnC +0, then BnC= 0 -ch alt 立

5. Suppose x is transcendental. Show that if k is a positive integer, then ** is transcendental. (Hint: Suppose x is algebraic and write an equation for xk. Show how this gives an equation for a.)
1. True or False? Prove your answer!Suppose A is a countably infinite set of real numbers that is bounded above, and let S = sup A. Then for every E > 0, the interval (S − E, S) contains a number in A and a number not in A.
the suprema and infima of the following sets: (a) {n €N | n² < 10}; (b) { | m, n = N}; n m+n n (c) {2+1 ne N}; (d) { | m, n € N with m + n ≤ 10}. m Compute, without proof,
3(i) and (ii) please!
1. For any real number r and any integer r #k, show that a) ()=(¹), and b) () = (-1)k (r+k-1). Hint: use the definition of () from Page 137. r-k
Is it always true that 1 - P[A] ≤ P[(AUB)c]? True False
6. If k divides |G| then 3H ≤ G such that |H|=k. *True *False
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3. If |F|q<∞ show: (a) There exists a prime p such that char F = p. (b) q = p some n. (c) a = a for all a € F. (d) If b € K is algebraic over F then b9" = b for some m > 0.