Let a and b be integers and consider (a) and (b) the ideals they
generate. Describe...
Let a and b be integers and consider (a) and (b) the ideals they
generate. Describe the intersection of (a) and (b), the product of
(a) and (b), the sum of (a) and (b) and the Ideal quotient
(aZ:bZ).
Ex 3. Consider the following definitions:
Definition: Let a and b be integers. A linear combination of a
and b is an expression of the form ax + by, where x and y are also
integers. Note that a linear combination of a and b is also an
integer.
Definition: Given two integers a and b we say that a divides b,
and we write a|b, if there exists an integer k such that b = ka.
Moreover, we write...
Let
I1, I2 be ideals of R and J1, J2 be ideals of S. Show that (I1 +
I2)^extension= I1^extension + I2^extension where I1, I2 are
contained in R
|^e
is defined as the extension of I to S: Let R and S be commutatuve
ring and f:R to S be a ring homomorphism. For each ideal I of R,
the ideal f(I)S of S generated by f(I) is the extension of I to
S.
a.) Prove the following: Lemma. Let a and b be integers. If both
a and b have the form 4k+1 (where k is an integer), then ab also
has the form 4k+1.
b.)The lemma from part a generalizes two products of integers of
the form 4k+1. State and prove the generalized lemma.
c.) Prove that any natural number of the form 4k+3 has a prime
factor of the form 4k+3.
Let a and b be integers. Recall that a pair of Bezout
coefficients for a and b is a pair of integers m, n ∈ Z such that
ma + nb = (a, b).
Prove that, for any fixed pair of integers a and b, there are
infinitely many pairs of Bezout coefficients.
Let a and b be integers which are not both zero.
(a) If c is an integer such that there exist integers x and y
with ax+by = c, prove that gcd(a, b) | c.
(b) If there exist integers x and y such that ax + by = 1,
explain why gcd(a, b) = 1.
(c) Let d = gcd(a,b), and write a = da′ and b = db′ for some
a′,b′ ∈ Z. Prove that gcd(a′,b′) = 1.
Let a,b be an element in the integers with a greater or equal to
1. Then there exist unique q, r in the integers such that b=aq+r
where z less than or equal r less than or equal a+(z-1). Prove the
Theorem.
Theorem 3.4. Let a and b be integers, not both zero, and suppose
that b = aq + r
for some integers q and r. Then gcd(b, a) = gcd(a, r).
a) Suppose that for some integer k > d, k | a and k | r. Show
that k | b also. Deduce that k is a common divisor of b and a.
b) Explain how part (a) contradicts the assumption that d =
gcd(b, a).
Let P be the uniform probability on the integers from 1 to 99.
Let B be the subset of numbers which have the digit 3. Let A be the
subset of even numbers. What is P(A), P(B)? What is P(A|B)?
P(B|A)?
Let a and b be positive integers, and let d be their greatest
common divisor. Prove that there are infinitely many integers x and
y such that ax+by = d. Next, given one particular solution x0 and
y0 of this equation, show how to find all the solutions.