4.4
J.A.Beachy
1
Review Problems
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
1. Use the Euclidean algorithm to nd gcd(x8 1, x6 1) in Q[x] and write it as a linear
combination of x8 1 and x6 1.
Solutio
3.8
3.8
J.A.Beachy
1
Cosets, Normal Subgroups, and Factor Groups
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
29. Dene : C R by (z) = |z|, for all z C .
(a) Show that is a group homomorphism.
Solution:
3.8
3.8
J.A.Beachy
1
Cosets, Normal Subgroups, and Factor Groups
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
The notion of a factor group is one of the most important concepts in abstract algebra.
To
4.4
4.4
J.A.Beachy
1
Polynomials over Z, Q, R, and C
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
21. Factor x5 10x4 + 24x3 + 9x2 33x 12 over Q.
Solution: The possible rational roots of f (x) = x5 10x4
3.7
3.7
J.A.Beachy
1
Homomorphisms
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
In Section 3.4 we introduced the concept of an isomorphism, and studied in detail what it
means for two groups to be isom
3.7
3.7
J.A.Beachy
1
Homomorphisms
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
21. Find all group homomorphisms from Z4 into Z10 .
Solution: As noted in Example 3.7.7, any group homomorphism from Zn i
4.4
4.4
J.A.Beachy
1
Polynomials over Z, Q, R, and C
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
Section 4.4 returns to the setting of high school algebra. The most important theorems
here are Eisenst
4.2
4.2
J.A.Beachy
1
Factors
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
This section introduces concepts for polynomials that model those for integers. The division algorithm for polynomials is simil
4.1
4.1
J.A.Beachy
1
Fields; Roots of Polynomials
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
25. Let F be a eld, and let f (x), g(x) F [x] be polynomials of degree less than n.
Assume that f (x) agre
4.2
4.2
J.A.Beachy
1
Factors
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
21. Over the eld of rational numbers, use the Euclidean algorithm to show that
2x3 2x2 3x + 1 and 2x2 x 2 are relatively prime.
4.3
J.A.Beachy
4.3
1
Existence of Roots
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
a b
a, b R
b a
phic to the eld of complex numbers.
25. Show that the eld F =
a b
b a
respects multiplication.
Soluti
4.3
4.3
J.A.Beachy
1
Existence of Roots
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
This section introduces congruences for polynomials, paralleling the development of congruence classes for integers.
4.1
4.1
J.A.Beachy
1
Fields; Roots of Polynomials
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
This section begins the study of elds in earnest. Besides the standard examples Q, R, and
C from high scho