Page 255 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Extension Field: Let F be an extension field of K and let u  F. If there exists a nonzero polynomial
f(x)  K[x] such that f(u) = 0, then u is said to be algebraic over K. If not, then u is said to be
transcendental over K.
In the above proof, the monic polynomial p(x) of minimal degree in K[x] such that p(u) = 0 is
called the minimal polynomial of u over K, and its degree is called the degree of u over K.

26.4 Review Questions

1. Find the splitting field over Q for the polynomial x + 4.
4
2. Let p be a prime number. Find the splitting fields for x – 1 over Q and over R.
p
3. Find the splitting field for x + x + 1 over Z .
3
2
4. Find the degree of the splitting field over Z for the polynomial (x + x + 1)(x + x + 1).
2
3
2
5. Let F be an extension field of K. Show that the set of all elements of F that are algebraic
over K is a subfield of F.
6. Let F be a field generated over the field K by u and v of relatively prime degrees m and n,
respectively, over K. Prove that [F : K] = mn.
7. Let F  E  K be extension fields. Show that if F is algebraic over E and E is algebraic over
K, then F is algebraic over K.
8. Let F  K be an extension field, with u  F. Show that if [K(u) : K] is an odd number, then
K(u ) = K(u).
2
9. Find the degree [F : Q], where F is the splitting field of the polynomial x – 11 over the field
3
Q of rational numbers.
10. Determine the splitting field over Q for x + 2.
4
11. Determine the splitting field over Q for x + x + 1.
2
4
12. Factor x – 1 over Z ; factor x – 1 over Z .
5
6
11
7
Answers: Self Assessment
1. (d) 2. (b) 3. (c) 4. (b)
26.5 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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