Last semester, a number of emails like this circulated:
Subject: Math Requirements
Dear James Stankewicz,
My name is Firstname Lastname and I am a graduate student at the University of Prestigious Institution studying for my ph d in math. I am taking a class and we are studying Field and Galois Theory and one of the requirements is to contact a graduate student from any other university and do an e-mail consultation with them regarding a problem pertaining to what we study. I would like to know if you would be able to help me fulfill this requirement. To do so, all you need do is answer the given problem to the best of your ability and then rate the difficulty of the problem for a Gradute Level Algebra Class on a scale from 1 – 10, 1 being the least difficult, and 10 being the most difficult. It would be more feasible if you could type your answer using the Latex software and e-mail it over as a pdf file, but it is not required. The problems go as follows:
1. Let S be the group of all permutations of F5. (Take note that no element of S is actually a field automorphism of F5, except the identity
permutation, since F5 has no nontrivial field automorphisms.) Let A be the
subgroup of even permutations. Let N be the subgroup of permutations of the form x|–> ax+b for a in F5^x, b in F5. Let D be the subgroup of
permutations of the form x|–> +/-x+b for b in F5. Let C be the subgroup of
permutations of the form x|–>x+b for b in F5.
(a) Show that N is the normalizer of C in S.
(b) Write down the cardinalities of the five groups S, A, N, D, and C, and determine all inclusion relations among them.
(c) Show that the only subgroups of S containing C are S, A, N, D, and C.
2. Let k be a field, let f(X) in k[X] be an irreducible quintic polynomial with five distinct roots, let K be the splitting field of f(X), and let G=Gal(K/k). Thus G can be identified with a subgroup of S5 via its action
on the roots of f(X).
(a) By making a suitable identification of the roots of f(X) with the
elements of F5, show that G is isomorphic to one of S, A, N, D, or C.
(b) Let k=Q, let p be a prime, and let f(X)=X^5 – p. Which group is G? What
is the fixed field of the subgroup C of G?
The letters S, A, N, D and C stand for “symmetric”, “alternating”, “normalizer”, “dihedral”, and “cyclic” , respectively
I got one of these from someone at a random university last month. Weird. Even if its legit I wouldn’t bother (but I doubt its legit, Firstname isn’t a first year and so probably isn’t taking a course on Field and Galois Theory..)
It was at this point that I contacted Firstname Lastname via his/her math department email and said that he/she ought to contact yahoo to get this email account closed down.
The name of the person has been protected because giving a name would only embarass the victim. If you see an email like this sent around, please make sure to contact the person whose name was used to get a response. No one deserves to have their name dragged through the mud like that!