III. Sample Problems
Consider f(x) = x4 - x2 - 2 E Q[x]. Find a splitting field K of
f(x) over Q. Compute [K : Q].
Suppose F UI K is a field extension of degree p, where p is a prime
number. Consider an element a E K \F.
(a) Find, with proof, the degree of a over F.
(b) Prove that K is a simple extension of F.
Let p be a prime number. Find the splitting field of x°P - 1 over F,
Assume that the characteristic of F is 0. Prove that any degree 2
field extension of F is normal.
Let G be the Galois group of x6 - 1 over Q. Prove that G is abelian
and identify G (that is, write G as a cyclic group or as a product
of cyclic groups).
These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
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