Download Mat2225, Number Theory, Homework 1

Mat2225, Number Theory, Homework 1
Hand in Tuesday, 16.8.2016 in the discussion session.
1. Use the Euclidean Algorithm inductively to find numbers a, b, c ∈ Z such that
30a + 105b + 42c = gcd(30, 105, 42)
Also, use the method of prime-factorization to confirm your answer for the greatest common divisor
gcd(30, 105, 42). Calculate lcm(30, 105, 42) as well.
2. Prove that for any a, b ∈ Z∗ := {n ∈ Z : n 6= 0} we find
{ma + nb : m, n ∈ Z} = {m · gcd(a, b) : m ∈ Z}
In words: the set of all numbers which are linear combinations of a, b in the integers are exactly the
multiples of the greatest common divisor of a and b.
3. Using polynomial division, one finds for any polynomial f (x) and non-zero polynomial g(x) two polynomials q(x) and r(x) such that
f (x) = g(x)q(x) + r(x)
and
deg(r) < deg(g)
where deg(f ) denotes the degree of the polynomial. (We refer to polynomials with real coefficients). This
leads to a Euclidean Algorithm to find a greatest common divisor where ’greatest’ refers to highest degree
of common divisors. Here, a polynomial function f (x) is a divisor of a polynomial function g(x) if one
finds a polynomial q(x) such that f (x)q(x) = g(x) (for all x ∈ R). As an example, x + 1 divides x2 − 1
because x2 − 1 = (x + 1)(x − 1) as a polynomial formula (i.e. it is true for all x).
Use the Euclidean Algorithm with polynomial division to find polynomials h(x) and k(x) such that
gcd(x2 − 2x − 3; x3 − 3x2 + x − 3) = h(x)(x2 − 2x − 3) + k(x)(x3 − 3x2 + x − 3) .
√
√
4. Consider the integral domain Z[ −6], these are all expressions of the form z = a + b −6, a, b ∈ Z. The
multiplication rule is
√
√
√
(a + b −6)(c + d −6) = ac − 6bd + −6(bc + ad) .
√
Assigning the norm |a + b −6|2 = a2 + 6b2√∈ Z+ we find the usual behavior |z1 z2 |2 =√|z1 |2 |z2 |2 . In
particular, a proper divisor w of z within√Z[ −6] (not unit and not associated
to z ∈ Z[ −6]) satisfies
√
|w|2 < |z|2 . Use this to show that 2, 3, ± −6 are irreducible within Z[ −6].
√
Find a√number in Z[ −6] with two different
factorizations into irreducible elements and show that
√
2, 3, ± −6 are not prime elements of Z[ −6].
√
In particular, there is no unique prime factorization within the integral domain Z[ −6].
1