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Robert B. Israel
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Joined: 24 Mar 2005
Posts: 2151 Posted: Sun Jul 16, 2006 8:00 pm    Post subject: Re: Three practice algebra qual problems Snis Pilbor <snispilbor@yahoo.com> wrote:
 Quote: Snis Pilbor wrote: Here are 3 practice algebra qual problems which stumped me. 1. Let S be a nonzero vector subspace of Q^n (Q the rational numbers). Let p be a prime number. Suppose that whenever (x_1,...,x_n) in S, then (p x_n,x_1,...,x_{n-1}) in S. In otherwords, S is invariant under the operation of multiplying the nth entry by p and then cycling entries. What I figured out: The map sending (x_1,...,x_n) to (p x_n,x_1,...,x_{n-1}) is a linear transformation whose matrix has row 1 equal to pe_n and row i equal to e_{i-1} for i=2,...,n, where e_i is the usual element of the standard basis. This matrix is obviously invertible and (less obviously) has min polynomial x^n-p. Maybe this would be useful for solving the problem... Argh, I was so wrapped up making sure the map in question was clear, I forgot to type the actual problem. The problem is, given these hypotheses, show S=Q^n.

Hint: if a subspace S of a vector space V is invariant under linear
transformation T, then T induces a linear transformation T^ of V/S
by T^ (v + S) = T v + S. What can you say about the minimal polynomial
of T^?

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada Achava Nakhash, the Lovin
science forum beginner

Joined: 14 Sep 2005
Posts: 35 Posted: Sun Jul 16, 2006 4:33 pm    Post subject: Re: Three practice algebra qual problems Snis Pilbor wrote:
 Quote: Here are 3 practice algebra qual problems which stumped me. 1. Let S be a nonzero vector subspace of Q^n (Q the rational numbers). Let p be a prime number. Suppose that whenever (x_1,...,x_n) in S, then (p x_n,x_1,...,x_{n-1}) in S. In otherwords, S is invariant under the operation of multiplying the nth entry by p and then cycling entries. What I figured out: The map sending (x_1,...,x_n) to (p x_n,x_1,...,x_{n-1}) is a linear transformation whose matrix has row 1 equal to pe_n and row i equal to e_{i-1} for i=2,...,n, where e_i is the usual element of the standard basis. This matrix is obviously invertible and (less obviously) has min polynomial x^n-p. Maybe this would be useful for solving the problem... 2. Let p be prime and consider g(x) in GF(p)[x] given by g(x)=x^p-x+a, where a in GF(p) is nonzero. Prove g(x) is irreducible over GF(p). What I figured out: g(x) obviously has no root in GF(p). It has distinct roots over a splitting field, and any field containing one root of g(x) contains all of them-- as a matter of fact if r is any root of g(x) in any field then r+1, r+2, ..., r+(p-1) are a full set of roots of g(x). 3. Give an example of a Unique Factorization Domain S and a nonzero prime ideal P in S such that S/P is not a field. What I figured out: if S is a PID and P is a nonzero prime ideal in S then S/P is a field. So we can restrict our attention to UFDs which arent PIDs. I seem to recall that these are extremely exotic, and I seem to remember that the "easiest" example of such a thing is Q(sqrt(-19)) but, even if this memory is accurate, this fact was highly nontrivial to prove, and anyway even if Q(sqrt(-19)) is a non-PID UFD, I have no idea how to pick P... Can anyone help out, these are not homework, they're practice qual problems I'm doing on my own so you guys are pretty much my only hope =) Snis Pilbor

For problem 3, first you are mistaken about Q(sqrt(-19). In fact any
algebraic number field is something called a Dedekind domain (unique
factorization of ideals into prime ideals or equivalent definitions to
be found in any algebraic number theory book.) Therefore any algebraic
number field that is a UFD is also a PID, I am not sure how what I
said proves this, but it is a highly interesting fact, anyway, and it
is true that UFD implies PID for algebraic number fields. Also ideals
in any algebraic number field can be generated by either one or two
elements.

Back to your problem. Since polynomials in a UFD form a UFD, K[x, y]
for any field K is a UFD that is not a PID. The ideal generated by x
is prime but not maximal, so this should answer your question.

For problem 2, I believe that you have done the hard part. I need to
find some air-conditioning before heat stroke carries me off. I will
return with the rest of this later today, I hope.

Regards,
Achava Snis Pilbor

Joined: 11 May 2005
Posts: 50 Posted: Sun Jul 16, 2006 2:16 pm    Post subject: Re: Three practice algebra qual problems Snis Pilbor wrote:
 Quote: Here are 3 practice algebra qual problems which stumped me. 1. Let S be a nonzero vector subspace of Q^n (Q the rational numbers). Let p be a prime number. Suppose that whenever (x_1,...,x_n) in S, then (p x_n,x_1,...,x_{n-1}) in S. In otherwords, S is invariant under the operation of multiplying the nth entry by p and then cycling entries. What I figured out: The map sending (x_1,...,x_n) to (p x_n,x_1,...,x_{n-1}) is a linear transformation whose matrix has row 1 equal to pe_n and row i equal to e_{i-1} for i=2,...,n, where e_i is the usual element of the standard basis. This matrix is obviously invertible and (less obviously) has min polynomial x^n-p. Maybe this would be useful for solving the problem...

Argh, I was so wrapped up making sure the map in question was clear, I
forgot to type the actual problem. The problem is, given these
hypotheses, show S=Q^n. karlfredrickson@gmail.com
science forum beginner

Joined: 15 Jul 2006
Posts: 2 Posted: Sun Jul 16, 2006 12:46 am    Post subject: Re: Three practice algebra qual problems Snis Pilbor wrote:

 Quote: 2. Let p be prime and consider g(x) in GF(p)[x] given by g(x)=x^p-x+a, where a in GF(p) is nonzero. Prove g(x) is irreducible over GF(p). What I figured out: g(x) obviously has no root in GF(p). It has distinct roots over a splitting field, and any field containing one root of g(x) contains all of them-- as a matter of fact if r is any root of g(x) in any field then r+1, r+2, ..., r+(p-1) are a full set of roots of g(x).

What you can do is assume that g(x) has a nontrivial factorization over
GF(p), say g(x)=f(x)h(x). If f and h have coefficients in GF(p) then
they will be fixed by the frobenius automorphism x -> x^p. But you can
also observe that the frobenius automorphism acts transitively on the
roots of g(x) in the splitting field, so if f or h has one root of g in
the splitting field, then it must have all the roots, which clearly
implies that one of f or g must be a constant.

 Quote: 3. Give an example of a Unique Factorization Domain S and a nonzero prime ideal P in S such that S/P is not a field. What I figured out: if S is a PID and P is a nonzero prime ideal in S then S/P is a field. So we can restrict our attention to UFDs which arent PIDs. I seem to recall that these are extremely exotic, and I seem to remember that the "easiest" example of such a thing is Q(sqrt(-19)) but, even if this memory is accurate, this fact was highly nontrivial to prove, and anyway even if Q(sqrt(-19)) is a non-PID UFD, I have no idea how to pick P...

If you want a number ring that is a non-PID UFD, then yes, you'll be
stuck with something kind of exotic and I'm not sure how you would
proceed. But it's a pretty standard fact that Z[x] (the ring of
integer polynomials) is UFD, and you should be able to find the needed
prime ideal in there fairly easily. Snis Pilbor

Joined: 11 May 2005
Posts: 50 Posted: Sat Jul 15, 2006 11:12 pm    Post subject: Three practice algebra qual problems Here are 3 practice algebra qual problems which stumped me. 1. Let S be a nonzero vector subspace of Q^n (Q the rational numbers). Let p be a prime number. Suppose that whenever (x_1,...,x_n) in S, then (p x_n,x_1,...,x_{n-1}) in S. In otherwords, S is invariant under the operation of multiplying the nth entry by p and then cycling entries. What I figured out: The map sending (x_1,...,x_n) to (p x_n,x_1,...,x_{n-1}) is a linear transformation whose matrix has row 1 equal to pe_n and row i equal to e_{i-1} for i=2,...,n, where e_i is the usual element of the standard basis. This matrix is obviously invertible and (less obviously) has min polynomial x^n-p. Maybe this would be useful for solving the problem... 2. Let p be prime and consider g(x) in GF(p)[x] given by g(x)=x^p-x+a, where a in GF(p) is nonzero. Prove g(x) is irreducible over GF(p). What I figured out: g(x) obviously has no root in GF(p). It has distinct roots over a splitting field, and any field containing one root of g(x) contains all of them-- as a matter of fact if r is any root of g(x) in any field then r+1, r+2, ..., r+(p-1) are a full set of roots of g(x). 3. Give an example of a Unique Factorization Domain S and a nonzero prime ideal P in S such that S/P is not a field. What I figured out: if S is a PID and P is a nonzero prime ideal in S then S/P is a field. So we can restrict our attention to UFDs which arent PIDs. I seem to recall that these are extremely exotic, and I seem to remember that the "easiest" example of such a thing is Q(sqrt(-19)) but, even if this memory is accurate, this fact was highly nontrivial to prove, and anyway even if Q(sqrt(-19)) is a non-PID UFD, I have no idea how to pick P... Can anyone help out, these are not homework, they're practice qual problems I'm doing on my own so you guys are pretty much my only hope =) Snis Pilbor  Display posts from previous: All Posts1 Day7 Days2 Weeks1 Month3 Months6 Months1 Year Oldest FirstNewest First
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