Abstract Algebra

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Abstract algebra in the Encyclopedia.


Basic Stuff

solution Explain the "null set."

Groups

proof Prove that the additive groups and are isomorphic for

proof Prove that if , then is abelian.

proof Prove that if is a group and are subgroups of , then is a group.

proof Prove that and , if and only if is a subgroup of .

proof Prove that a group homomorphism maps the identity to the identity and inverses to inverses.

proof Prove that the kernel of a homomorphism from a group is a subgroup of that group.

solution Define actions, centralizers, normalizers, stabilizers, and centers.

proof Prove that if and then the following are equivalent:
   (a) which means is a generator of .
   (b) i.e. and are relatively prime.
   (c) such that .

solution Let and belong to the group . If and , where and are relatively prime, show that and that .

proof Let be any subgroup of the group . The set of left cosets of in form a partition of . Furthermore, for all if and only if and in particular, if and only if and are representatives of the same coset.

proof If is a subgroup of and is a subgroup of , prove is a subgroup of .

proof Let and , each of prime order , be subgroups of a group . If , prove .

proof If and are prime, show every proper subgroup of a group of order is cyclic.

proof Let be a group such that ( for all . Prove is abelian.

proof Let be a group such that for all and for three consecutive integers . Prove is abelian.

proof Prove that a group of order 56 has a normal Sylow p-subgroup for some prime dividing its order.

proof Prove that a group of order 312 has a normal Sylow p-subgroup.

Groups- facts and examples

  • Addition of residue classes in is associative.
  • Multiplication of residue classes in is associative.
  • A finite group is abelian if and only if its group table is a symmetric matrix.
  • .
  • If then .
  • If then if and only if .
  • If , then is abelian.
  • is abelian if and only if both and are abelian.

Dihedral Groups

  • If and is not a power of , then .
  • Every element of which is not a power of is of order 2.

Symmetric Groups

  • is a non-abelian group for all .
  • Disjoint cycles commute.
  • The order of a permutation is the l.c.m. of the lengths of the cycles in its cycle decomposition.
  • An element has order 2 in if and only if its cycle decomposition is a product of commuting 2-cycles.
  • Let be a prime. Show that an element has order in if and only if its cycle decomposition is a product of commuting p-cycles.
  • If then the number of -cycles in is given by .


Matrix Groups

  • is an matrix with entries from and nonzero determinant.
  • is non-abelian.

Homomorphisms and Isomorphisms

  • is a homomorphism if . This implies and .
  • The exponential map defined by is an isomorphism from to .


Rings

solution In the ring , prove the following:

(a) if and only if

(b) where . You need to show both equalities. Note that this implies that any ideal in the ring is principal.

solution Determine whether is irreducible over or not.

solution Find a g.c.d. of and in by the Euclidean algorithm.

solution Show that is irreducible but not a prime in .

solution Show that is a maximal ideal of .

solution (Dis)prove: Let be a commutative ring with more than one element. If for every nonzero element of , we have , then is a field.

solution (Dis)prove: as rings, where is prime.

solution Show that an integral domain with a descending chain condition (if is a descending chain of ideals, then there exists such that ) is a field.

proof Let be an integral domain. Suppose that existence of factorizations holds in . Prove that is a unique factorization domain if and only if every irreducible element is prime.

solution Prove: If is an integral domain and s.t. then .

solution Let be a nilpotent element of the commutative ring . Let for minimal . Prove that is either zero or a zero divisor.

solution Let be a nilpotent element of the commutative ring . Let for minimal . Prove that is nilpotent for all .

solution Let be a nilpotent element of the commutative ring . Let for minimal . Deduce that the sum of nilpotent element and a unit is a unit.
solution Prove that the center of a ring is a subring containing the identity.

solution Prove that the center of a division ring is a field.

Chinese Remainder Theorem

Statement of theorem: Let be a ring with identity .

Let be ideals of . The map defined by is a ring homomorphism with kernel . If for each with the ideals and are comaximal, then this map is surjective and , so


solution Let be integers which are coprime to each other.

(a) Show that the Chinese Remainder Theorem implies that for any there is a solution to the simultaneous congruences

and that the solution is unique mod .



(b) Let be the quotient of by . Prove that the solution in (a) is given by

.



(c) Solve the simultaneous systems of congruences

and

.


Euclidean Domains

solution Define a Euclidean Domain.

solution Let R be a Euclidean Domain with a function . Prove that
(a)
(b)
(c) Use (b) to determine and

Fields

solution Calculate the splitting field of over . What is ?

solution Prove that the polynomial is irreducible:

solution Prove that the polynomial is irreducible:

solution Show that the splitting field of is a simple extension of .