301 Abstract Algebra
2:00 - 2:50 MWF
Hentzel 432 Carver
hentzel@iastate.edu
Web Page   http://www.math.iastate.edu/hentzel/class.301

Text: A First Course in Abstract Algebra, sixth Edition
by John B. Fraleigh
Course: Chapter 1,2,3,4,5

Monday, November 26.  

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Practice test.

The Euclidean property for the integers is:

Given a and b, b =/= 0, then there is a q and

r such that a = bq+r and 0 <= r < |b|. 

In this test you may assume that the integers have the 

Euclidean property.

1. Prove that in the integers we have factorization into

irreducibles.





2.a  What is the greatest common divisor property?

2.b  Prove that the integers have greatest common divisors.







3.a  What is the definition of irreducible and prime? 

3.b  Prove in the integers that every irreducible is a prime.






4. Prove that in the integers we have unique factorization into

irreducibles.












5.  If a < 0, prove that the function n(x+y Sqrt[a]) = x^2 - a y^2 

is multiplicative on Z[Sqrt[a]].







6.  Prove that in Z[Sqrt[-5]] we do not have unique factorization

into irreducibles.






7.  In the Gaussian Integers,

(a) Find the Greatest Common Divisor of (8+2 i) and (4 -3 i). 

(b) Find Gaussian Integers x and y such that

        x(8+2 i) + y(4-3 i) = GCD(8+2 i, 4-3 i).





8.  Prove that if you have a multiplicative norm n with the property that

n(a) = 1 <===> a is a unit, then you have factorization into 

irreducibles.





9.  Find q and r in Z[Sqrt[3]] such that

523+ 8 Sqrt[3] = (43 + 6 Sqrt[3])q + r and n(r) < n(43+6 Sqrt[3]).

n(a+b Sqrt[3]) = | a^2 - 3 b^2 |.

10. In the even integers 2 Z, 

(a) Which elements are irreducibles? 

(b) Which elements are primes?