Page 296 - DMTH403_ABSTRACT_ALGEBRA

Unit 31: The Galois Group of a Polynomial

Galois Group of the Equation: Let K be a field, let f(x) be a polynomial in K[x], and let F be a Notes
splitting field for f(x) over K. Then Gal(F/K) is called the Galois group of f(x) over K, or the
Galois group of the equation f(x) = 0 over K.

31.8 Review Questions

1. Let p be prime. Prove that there exists a polynomial f(x)  Q[x] of degree p with Galois

group isomorphic to S . Conclude that for each prime p with p  5 there exists a polynomial
p
of degree p that is not solvable by radicals.
2. Let p be a prime and Z (t) be the field of rational functions over Z . Prove that f(x) = x – t
p
p
p
is an irreducible polynomial in Zp(t)[x]. Show that f(x) is not separable.
3. Let E be an extension field of F. Suppose that K and L are two intermediate fields. If there
exists an element   G(E/F) such that (K) = L, then K and L are said to be conjugate fields.
Prove that K and L are conjugate if and only if G(E/K) and G(E/L) are conjugate subgroups
of G(E/F).
4. Let   Aut(). If a is a positive real number, show that (a) > 0.
5. Let K be the splitting field of x + x + 1   [x]. Prove or disprove that K is an extension by
3
2
2
radicals.
6. Let F be a field such that char F  2. Prove that the splitting field of f(x) = ax + bx + c is
2
),
F(  where a = b – 4ac.
2
7. Prove or disprove: Two different subgroups of a Galois group will have different fixed
fields.
8. Let K be the splitting field of a polynomial over F. If E is a field extension of F contained
in K and [E : F] = 2, then E is the splitting field of some polynomial in F[x].

Answers: Self Assessment

1. (b) 2. (b) 3. (c) 4. (d) 5. (a)

31.9 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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