This set is due February 6 at the beginning of lecture. Consult the syllabus for details on the homework policy.

1. a. Complete the proof by induction that if are integers and , then for all integers .

b. This result allows us to give a nice proof of the following fact: Let be a natural number and let be a positive integer. If the -th root of , , is rational, then it is in fact an integer. (The book gives a proof of a weaker fact.) Prove this result as follows: First verify that if and , then . Show that any fraction with integers, can be reduced so . Assume that is rational, say . Then also . Express this last fraction as a rational number in terms of . Use that for all and the general remarks mentioned above, to show that is in fact an integer.

2. Show by induction that for all integers there is a polynomial with rational coefficients, of degree and leading coefficient , such that for all integers , we have There are many ways to prove this result. Here is one possible suggestion: Consider .

3. Euclidean algorithm. We can compute the gcd of two integers not both zero, as follows; this method comes from Euclid and is probably the earliest recorded algorithm. Fist, we may assume that are positive, since , and also we may assume that , so . Now define a sequence of natural numbers as follows:

, .

Given , if , then .

Otherwise, , and we can use the division algorithm to find unique integers with such that . Set .

Let be the set of those that are strictly positive. This set has a least element, say . By the way the algorithm is designed, this means that .

Show that , and that we can find from the algorithm, integers such that .

(If the description above confuses you, it may be useful to see the example in the book.)

4. Assume that the application of the algorithm, starting with positive integers , takes steps. [For example, if and , the algorithm gives:

Step 1: , so .

Step 2: , so .

Step 3: , so , and . Here, ]

Show that , where the numbers are the Fibonacci numbers, see Exercises 15-22 in Chapter 1 of the book.

5. Extra credit problem. With as in the previous exercise, let be the number of digits of (written in base 10). Show that

This entry was posted on Monday, January 26th, 2009 at 2:24 pm and is filed under 305: Abstract Algebra I. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

[…] Homework 1, due February 6, at the beginning of lecture. Possibly related posts: (automatically generated)What Lil’ Ones Are Reading: Two Christmas Stories […]

Perhaps this is a stupid questions, but what does the notation (a,b) = 1 mean in the first problem? Is it a function that I have missed somewhere? Is it equivalent to a = b = 1, so that we are to prove that a = a^n = 1 and b = b = 1 for all integers n >= 1? I’m a little lost…

Hi. The notations and mean the same thing: The greatest common divisor of and .
[I agree there is a little ambiguity in using this way, but it is standard in number theory.]

[…] Homework 1, due February 6, at the beginning of lecture. Possibly related posts: (automatically generated)What Lil’ Ones Are Reading: Two Christmas Stories […]

Perhaps this is a stupid questions, but what does the notation (a,b) = 1 mean in the first problem? Is it a function that I have missed somewhere? Is it equivalent to a = b = 1, so that we are to prove that a = a^n = 1 and b = b = 1 for all integers n >= 1? I’m a little lost…

Hi. The notations and mean the same thing: The greatest common divisor of and .

[I agree there is a little ambiguity in using this way, but it is standard in number theory.]

[…] Homework 1, due February 6, at the beginning of lecture. [This homework was not graded.] […]