Show that $ and %$%^-1 have the same parity for all % and $ in Sn. (Sn is the symmetric group of degree n)
Permutations : Disjoint Transpositions
If $ belongs to Sn (where Sn is the symmetric group of degree n), show that $^2 = % if and only if $ is a product of disjoint transpositions.
Define the sign of a permutation $ to be: sgn $ = 1 if $ is even. -1 if $ is odd. Prove that sgn($%) = sgn$sgn% for all $ and % in Sn.
Integral Domains Fields and Subfields
Problem: Note: Q is rational numbers, R is real numbers , sqrt() means square root Show that Q(sqrt(2)) is the smallest subfield of R that contains sqrt(2).
Integral Domains and Fields : Embedding Theorem
Problem: NOte: C is set containment If R is an integral domain, show that the field of quotients Q in the Embedding Theorem is the smallest field contianing R in the following sense: If R C F, where F is a field, show that F has a subfield K such that R C K and K is isomorphic to Q.
Problem: Note: | | is trying to denote a matrix If R = |S S| |0 S| and A = |0 S| |0 0| , S and ring, show that A is an ideal of R and describe the cosets in R/A
Please if you are going to solve this problem please be very accurate!! (and careful!!). I have used brainmass before and have recieved incorrect answers on mulitple occations (answers that initially seemed correct) and it has hindered my studying. I have decided to give it another chance. With that said, thank you very much for ...continues
Ideals and Factor Rings : Annihilator
Problem: If X is contained in R is a nonempty subset of a commutative ring R, define the annihilator of X by ann(X) = { a belonging to R | ax=0 for all x belonging to X} Show that X is contiained in ann[ann(X)] AND Show that ann(X) = ann{ann[ann(X)]}
Ideals and Factor Rings : Prime Ideal
Problem: Let R be a commutative ring. Show that every maximal ideal of R is prime and if R is finite, show that every prime ideal is maximal. ALSO is every prime ideal of Z(integers) maximal? Why?
Ideals and Factor Rings : Locality
Problem: A ring R is called a local ring if the set J(R) of nonunits in R forms an ideal. If p is a prime, show that Z(p) = {n/m belonging to Q | p does not divide m } is local. Describe J(Z(p))
