Exercises 6.7 Additional Exercises: The Euclidean Algorithm
1.
Find the greatest common divisor of 471 and 564 using the Euclidean Algorithm and then find integers \(r\) and \(s\) such that \(\gcd(471,564) = 471r+564s\text{.}\)
2.
Find a single integer \(n\) such that the ideal \(\langle n \rangle\) is the smallest ideal in \(\Z\) containing both \(471\) and \(564\text{.}\)
3.
In the quotient ring \(\Z/\langle 564 \rangle\text{,}\) find an element \(a + \langle 564\rangle\) such that \((a+\langle 564\rangle)(471 + \langle 564\rangle) = 3 + \langle 564 \rangle\text{.}\)
4.
Is \(471 + \langle 564\rangle\) a unit in \(\Z/\langle 564\rangle\text{?}\) Explain.
5.
In \(\Q[x]\text{,}\) find the greatest common divisor of the polynomials \(a(x) = x^3 + 1\) and \(b(x) = x^4 + x^3 + 2x^2 + x - 1\text{.}\) Then express the gcd as a combination of the two polynomials (as in Bezout's lemma).
6.
Find the greatest common divisor of \(x^{24}-1\) and \(x^{15}-1\) in \(\Q[x]\text{,}\) and then express the gcd as a combination of the two polynomials.
7.
Find a coset \(a(x) + \langle x^{24}-1\rangle\) of \(\Q[x]/\langle x^{24}-1\rangle\) such that \((a(x) + \langle x^{24}-1\rangle)(x^{15}-1) + \langle x^{24}-1\rangle) = x^3-1 + \langle x^{24}-1\rangle\text{.}\)