Page 250 - DMTH403_ABSTRACT_ALGEBRA

Unit 25: Roots of a Polynomial

25.4 Review Questions Notes

1. Let F be a field and f(x)  F[x] with deg f(x)  1. Let a  F. Show that f(x) is divisible by
x – a iff f(a) = 0.
2. Find the roots of the following polynomials, along with their multiplicity.

1 5
2
(a) f(x)  x  x 3 Q[x] (b) f(x)  x  x 1 Z [x]
2




2 2 3
(c) f(x)  x  2x  2x  1 Z [x]
4
3

5
3. Let F be a field and a  F. Define a function
 : F[x]  F : f(f(x)) = f(x)
This function is the evaluation at a.
Show that
(a) f is an onto ring homomorphism.
(b) f (b) = b  b  F.
(c) Ker f = <x – a>
So, what does the Fundamental Theorem of Homomorphism say in this case?

4. Let p be a prime number. Consider x p 1  1 Z [x]. Use the fact that Zp is a group of order


p
p to show that every non-zero element of Zp is a root of x – 1. Thus, show that x – 1 =
p-1
p-1
(x 1)(x 2)...(x p 1).




5. The polynomial x + 4 can be factored into linear factors in Z [x]. Find this factorisation.
4
5
Answers: Self Assessment
1. (c) 2. (c) 3. (c) 4. (a) 5. (d)
25.5 Further Readings

Books Dan Saracino: Abstract Algebra; A First Course.
Mitchell and Mitchell: An Introduction to Abstract Algebra.
John B. Fraleigh: An Introduction to Abstract Algebra (Relevant Portion).

Online links www.jmilne.org/math/CourseNotes/
www.math.niu.edu
www.maths.tcd.ie/
archives.math.utk.edu

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