Test 2. Monday, November 12, 2018

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Kristian Armstrong
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1 Test 2 Monday, November 12, 2018 Instructions. The only aids allowed are a hand-held calculator and one cheat sheet, i.e. an sheet with information written on one side in your own handwriting. No cell phones are permitted (in particular, a cell phone may not be used as a calculator). Answer as clearly and precisely as possible. Clarity is required for full credit! Time permitted: 50 minutes. 1. (15 points) Let S be the set of all real numbers whose decimal expansion has the form ±. a b c d e f a b c d e f a b c... with an arbitrary finite string of decimal digits before the decimal point, and six decimal digits repeating forever after the decimal point. Thus, for example, and similarly, = 1 7 S; = 1 3 S. What is the least common denominator of all the elements of S? (In other words, find the smallest positive integer m such that ms Z, where ms = {ms : s S}).

2 2. (15 points) Give explicit examples of each of the following: (a) A polynomial f(x) Z[x] of degree three having no rational roots; (b) a polynomial f(x) Q[x] of degree four having no real roots; and (c) a polynomial f(x) Z[x] of degree three having 3 2, 1+ 2 and 1 2 as its roots.

3 3. (20 points) Consider two polynomials f(x), g(x) C[x], not both zero, and let d(x) = gcd(f(x), g(x)). (a) If d(x) = 1, show that f(x) and g(x) have no roots in common (i.e. there is no complex number r satisfying f(r) = g(r) = 0). (b) Show that conversely, if f(x) and g(x) have no roots in common, then d(x) = 1.

4 4. (20 points) Multiple choice: Circle A,B,C or D to indicate the best completion of each sentence. (a) The proof that 2 is irrational A. requires a knowledge of the complex numbers. B. relies on the fact that the decimal digits of 2 continue forever without repetition. C. makes essential use of Euclid s Lemma. D. makes essential use of Fermat s Little Theorem. (b) The complex number system C A. consists of all numbers constructible from the previously known number systems Z, Q and R. B. is algebraically closed. C. contains infinitely many elements between z 1 and z 2, for any two distinct complex numbers z 1 < z 2. D. is just the case n = 2 of a natural sequence of fields R C H O D of dimension n = 1, 2, 3,.... (c) The real number system R A. is defined as the set of numbers in natural one-to-one correspondence with points on a physical line which extends forever in both directions. B. contains elements of every possible size: zero, finite, infinite and infinitely small. C. contains rational and irrational values, the distinction between rational and irrational depending on the choice of base of representation (with base 10 being the most popular choice in this age and culture). D. extends the field Q of rational numbers; and the definition of R utilizes and builds upon an understanding of Q. (d) If f(x) Q[x] is a polynomial of degree 5 then A. f(x) has at least one root in Q. B. f(x) has at least one root in R. C. f(x) must have four irrational roots in R. D. f(x) must have four non-real roots in C. (e) Given two complex numbers z, w C, A. The value of z w C is well-defined. B. The real part of z equals the imaginary part of iz. C. z + w = z + w. D. zw = zw.

5 5. (30 points) Answer TRUE or FALSE to each of the following statements. (a) If a, b R are irrational, then ab must also be irrational. (b) For every positive integer n, there exists an irreducible polynomial f(x) C[x] of degree n. (c) If z is a nonzero complex number, then z 1 = 1 z 2 z. (d) Every nonzero complex number has the form e z for some z C. (e) Every complex number z satisfies e z = e z. (f) If p is prime, then the digits in the decimal expansion of 1 p repeat with period at most p 1. (g) The real number e = k=0 1 k! is irrational. (h) There exists a positive real number c which is less than 1 n integer n. for every positive (i) For every positive integer n, there exists a real number c such that 0 < c < 1 n. (j) For every real number r and every positive integer n, there exists x Q such that x r < 1 n.

CHAPTER 1 REAL NUMBERS KEY POINTS

CHAPTER 1 REAL NUMBERS KEY POINTS CHAPTER 1 REAL NUMBERS 1. Euclid s division lemma : KEY POINTS For given positive integers a and b there exist unique whole numbers q and r satisfying the relation a = bq + r, 0 r < b. 2. Euclid s division