GALOIS THEORY COMPREHENSIVE TEST
PAPER QUESTIONS AND SOLUTIONS
GRADED APLUS
●● Cyclotomic Polynomial
Answer: The general factors of t^n - 1. They take the form Φn(t).
●● Primitive Root of Unity
Answer: A root where its powers generate all the roots before returning
to 1. For example, ζ8k is primitive ⟺ gcd(k, 8) = 1.
●● Galois Theorem
Answer: The polynomial f in Q is solvable by radicals iff Gal(f) is a
solvable group.
●● Sym(X)
Answer: Otherwise known as Sn, the symmetric group is all
permutations of n objects.
●● Transposition
Answer: The interchanging of two elements.
, ●● Even Permutation
Answer: The product of an even number of transpositions.
●● Alternating Group
Answer: The group consisting of all even permutations in Sym(X).
●● Action
Answer: A function G x X → X, written (g, x) → gx where:
- (gh)x = g(hx)
- 1x = x
●● Faithful Action
Answer: The corresponding homomorphism is injective.
●● Homomorphism
Answer: A function satisfying:
- φ(r + r ′ ) = φ(r) + φ(r ′ )
- φ(rr′ ) = φ(r) * φ(r ′ )
- φ(1) = 1.
●● Subring
Answer: A subset S of a ring R, satisfying:
PAPER QUESTIONS AND SOLUTIONS
GRADED APLUS
●● Cyclotomic Polynomial
Answer: The general factors of t^n - 1. They take the form Φn(t).
●● Primitive Root of Unity
Answer: A root where its powers generate all the roots before returning
to 1. For example, ζ8k is primitive ⟺ gcd(k, 8) = 1.
●● Galois Theorem
Answer: The polynomial f in Q is solvable by radicals iff Gal(f) is a
solvable group.
●● Sym(X)
Answer: Otherwise known as Sn, the symmetric group is all
permutations of n objects.
●● Transposition
Answer: The interchanging of two elements.
, ●● Even Permutation
Answer: The product of an even number of transpositions.
●● Alternating Group
Answer: The group consisting of all even permutations in Sym(X).
●● Action
Answer: A function G x X → X, written (g, x) → gx where:
- (gh)x = g(hx)
- 1x = x
●● Faithful Action
Answer: The corresponding homomorphism is injective.
●● Homomorphism
Answer: A function satisfying:
- φ(r + r ′ ) = φ(r) + φ(r ′ )
- φ(rr′ ) = φ(r) * φ(r ′ )
- φ(1) = 1.
●● Subring
Answer: A subset S of a ring R, satisfying:
