(f) Let T E R. Suppose (T) = p, where p is a rational prime (i.e., a prime integer).Show that is prime in Rπ(g) Show that every prime π E R divides a rational prime p.(h) Show that a rational prime p either factors as p = uл, where u = 1 or − 1 andπ and are prime in R, or p remains prime in R. Let R = Z[√2] = {a+b√2a, b e Z} andThen R is a ring, F is a field, and RC F CRDefine the norm on F by setting v(a+b√√/2) ==

R=ℤ[√2]=a+b2:|a,b∈ℤ F=ℚ[2]=a+b2:|a,b∈ℚ Define the norm on F by setting v(a+b2)=|a2-2b2| α=a+b2…

Answered: (f) Let T E R. Suppose (T) = p, where p…

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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3. Let f = ax? + bx + c, for a, b, c real numbers and a + 0. Prove that f is a prime in R[x] if and only if b? 4ас3D D 2 0 then 4ac < 0. Prove that if b2 - -b + VD -b * - 2a 2a
6. Let | En+1 = Cen|En-1. The ansatz |En+1 = ken|P results in √3+1 2 O √3-1 2 Op= √5 +1 2 O √5-1 2 7. Which of the following is a sixth root of unity? O i O-i 01/1/0 + NICO 2.
(a) Prove for every pair p, q of distinct primes that Spq is irrational. (b) Prove for every pair p, q of distinct primes that p+ vq is irrational.
2. Consider the function y(x) = 5x over the interval [0, 10]. (a) If the points A(0, y(0)), B(5, y(5)), and C(10, y(10)) are on the curve represented by the above function, find the slope of the lines (secants) AB, and AC, respectively. (b) If x represents time and y represents the displacement of a particle moving along a line, what is the physical meaning of your results to part (a)? (c) Now consider a point D(10 – h, y(10 – h)) on the curve represented by the given function, find the slope of the line DC. (d) As h → 0, what is the limiting value of your result to part (c)? (e) What is its physical meaning of your result to part (d) if again x represents time and y represents the displacement?
3. Recall that a rational number r EQ is a number of the form r = where a, b & Z, b +0. Decide whether each of the following claims is true or false. Then prove your answers. Claim 1. There is a positive rational number smaller than any other positive rational number. Claim 2. For any r € Q with r = 0, there exists m € Z for which rm € N = {1,2,3, ...}. 4. Suppose that r € Q, r‡ 0, and that x € R\Q (x is irrational). Prove that rx is also irrational. 5. Prove that for any two nonnegative real numbers, their arithmetic mean is always greater than or equal to their geometric mean. Meaning, prove that for any x,y ER where x, y ≥ 0, x+y 2 Σvry. Moreover, prove that there is equality if and only if x = y.
prove that the below is not rational

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(f) Let T E R. Suppose (T) = p, where p is a rational prime (i.e., a prime integer). Show that is prime in R (g) Show that every prime T E R divides a rational prime p. (h) Show that a rational prime p either factors as p = uл, where u = 1 or -1 and π and are prime in R, or p remains prime in R.