Homework Statement
Let R be an integral domain and suppose that R[x] is a principal ideal domain. Show that R is a field.
Homework Equations
I don't know where to start, I'm not familiar with this material. I was browsing through an abstract algebra book and found this. Would like...
Currently I am reviewing basic algebra, trigonometry and I will also be starting calculus this fall semester...
I enjoy reading about math and I wanted to know what abstract algebra is? Would this be to difficult to read seeing that I am only starting calculus?
If so what other types of...
Homework Statement
Consider \frac{\mathbb Z_2[X]}{X^2+1}, is this ring isomorphic to \mathbb Z_2 \oplus \mathbb Z_2, \mathbb Z_4 or \mathbb F_4 or to none of these?
Homework Equations
/
The Attempt at a Solution
- \mathbb F_4 No, because \mathbb Z_2[X] is a principle ideal domain...
Hello,
I just took ordinary diff eq and I've had calc III and linear algebra, but I'm worried about taking Modern Algebra or Real Analysis next semester because I have no experience writing proofs. The linear algebra class was all computation on tests and homework (we did see some proofs on...
The .pdf can be ignored.
Let A + B = (A - B) U (B - A) also known as the symmetric difference.
1. Look for the identity and let e be the identity element
A + e = A
(A - e) U (e - A) = A
Now there are two cases:
1. (A - e) = A
This equation can be interpreted as removing from A all elements...
Homework Statement
Let G be a group with pk elements, where p is a prime number and k is greater than or equal to 1. Prove that G has a subgroup of order p. The Attempt at a Solution
I attempted to prove this by showing that the conditions for a set to be subgroup form a subgroup of order p...
I really feel dissapointed in myself that I didn't perform as well as I wanted last semester. I took Modern Algebra I and Geometry. The Geometry class covered Euclidean and non-Euclidean geometries. I bombed the final but earned an overall of a B+ because of a 90-something percentile homework...
I've been studying cryptography and I found out that AES uses Galois Fields. I was therefore wondering where else does abstract algebra pop-up for real world use?
What is the absolute best abstract algebra book for graduate students? I was wanting a book that covers algebra in the most comprehensive manner possible, at about the level of Hungerford's Algebra. I was wondering if Carstensen's Abstract Algebra in the Sigma Series in Pure Mathematics is a...
Homework Statement
Let D = Z[sqrt(10)], and let P be the ideal (2,sqrt(10)) 10). Prove that P is a prime
ideal of D.
Homework Equations
The Attempt at a Solution
Not sure where to start. I think elements are of the for a+b*sqrt(10), a,b integers.
Any hints as to what to do next?
I have started to write Abstract Algebra notes as I am learning them, and typing them with LaTex afterwards. I have just done a bit but I want some of you to help and see if I have got any thing wrong (having the wrong concept in your mind can have terrible consequences) or anything else to make...
Homework Statement
Does anyone know what left and right translations are good for?
\begin{cases} R_{a}g=ga\\L_{a}g=ag\end{cases} with a,g\in G and G is a Lie group
How can we interpret these relations in the easiest way like we try to explain it to a student which if not familiar with...
I was wondering if anyone knew any links on the Internet that help to explain abstract algebra and maybe works through some problems as well. Thank you in advance
Homework Statement
Show that x^2\,+\,x can be factored in two ways in \mathbb{Z}_6[x] as the product of nonconstant polynomials that are not units.Homework Equations
Theorem 4.8
Let R be an integral domain. then f(x) is a unit in R[x] if and only if f(x) is a constant polynomial that is a...
Homework Statement
Suppose |G| = pqr where p, q, and r are distinct primes. If H is a subgroup of G and K is a subgroup of G with |H| = pq and |K| = qr, then |H intersect K| = q.
Homework Equations
NA
The Attempt at a Solution
I have so far:
Let a be an element of H intersect K...
I was wondering if anyone could give me any links or an introduction to abstract algebra. I know that abstract algebra is a tough concept to understand (at least for some people, but it varies from person to person). If anyone could help with the basics of it would be greatly appreciated.
Homework Statement
Let G be a nonempty finite set closed under an associative operation such that both the left and right cancellation laws hold. Show that G under this operation is a group.
Homework Equations
My book defines the left and right cancellation laws as :
"For any a,b in...
1. Homework Statement [/b]
The set of positive real numbers, R+, is a group under normal multiplication. The set of real
numbers, R, is a group under normal addition. For the sake of clarity, we'll call these groups G and H respectively.
Prove that G is isomorphic to H under the isomorphism...
Hello Experts,
I can't find the proof of this theorems please help me:
Given that there is a commutative ring R and 2 ideals I and J, also given that I is included in J
I need to prove
1) radical of I is in radical of J
2) radical of radical of ideal I = radical of ideal I...
Hi Folks.
I was hoping to pick the brains of some of the mathematicians and mathematically inclined on this site.
I'm very interested in how mathematicians think about abstract objects that don't seem to be grounded in anything concrete. In particular, how do mathematicians think to...
Homework Statement
from Algebra by Michael Artin, chapter 2, question 5 under section 2(subgroups)
An nth root of unity is a complex number z such that z^n =1. Prove that the nth roots of unity form a cyclic subgroup of C^(x) (the complex numbers under multiplication) of order n...
I was wondering if one wanted to pursue learning more about cryptography which of these classes would be the most important? Number theory of abstract algebra?
Homework Statement
a) Show that there is exactly one maximal ideal in Z_8 and in Z_9.
b) Show that Z_10 and Z_15 have more than one maximal ideal.
Homework Equations
I know a maximal ideal is one that is not contained within any other ideal (except for the ring itself)
By...
Homework Statement
(This is an example of a group in my text).
An integer 'a' has a multiplicative inverse modulo n iff 'a' and 'n' are relatively prime. So for each n > 1, we define U(n) to be the set of all positive integers less than 'n' and relatively prime to 'n'. Then U(n) is a group...
Homework Statement
For any integer n>2, show that there are at least two elements in U(n) that satisfy x^2 = 1.
Homework Equations
None
The Attempt at a Solution
If the definition of the group U(n) is "the set of all positive integers less than n and relatively prime to n" then the...
Homework Statement
Does the rule g*x = xg^-1 define an operation of G on G?
Homework Equations
The Attempt at a Solution
I don't even know what this means. Could someone just tell me what it means for a rule to define an operation of one group on itself? I should be able to figure...
I'm a physics undergrad and doing some undergrad study on QFT, and I found that Lie algebra is often invoked in texts, so I decide to take a Lie algebra this sem but I've not taken any abstract algebra course before.The first day's class really beats me because the lecturer used many concepts...
Homework Statement
Find the number of elements in the ring Z_5[x]/I, where I is a) the ideal generated by x^4+4, and b) where I is the ideal generated by x^4+4 and x^2+3x+1.
Homework Equations
Can't think of any.
The Attempt at a Solution
I started by finding the zeros of the...
Would this be a good thing to take? More specifically, will a introduction to this shed light on/put on more solid ground many of the techniques/organizations in physics that are presented as "tricks"?
I just want to be sure it will be worth it, since i'll be taking it alongside...
I'm taking this course "abstract algebra" at university and I've been given some homework questions. I was able to solve all of them but one. And it would be great if anyone could help me with this.
The question is like this:
"If all cyclic subgroups of G are normal, then show that all...
Homework Statement
List the elements of the cyclic subgroup of S_6 generated by
f = \left(\begin{array}{llllll}
1 & 2 & 3 & 4 & 5 & 6\\
2 & 3 & 4 & 1 & 6 & 5\\
\end{array}\right)Homework Equations
The Attempt at a Solution
I really do not understand what the elements of a permutation really...
I am a senior student double majoring in computer science and mathematics with the intention
of getting a p.h.d in theoretical computer science(either computational complexity or applied discrete mathematics). for the upcoming winter semester I can take 1 math course. The ones that are related...
Homework Statement
Let A be a a square n*n matrix. Prove that A^-1 has only integer enteries if and only if the determinant of A is + or -1.
Homework Equations
general knowledge of determinants
The Attempt at a Solution
Proof:
=>
Suppose that det(A) = 1 (without losing...
Homework Statement
Use the First Isomorphism Theorem to show that Q[x]/(x^3-3) is isomorphic to {a+b*sqrt(3)}
Homework Equations
First Isomorphism Theorem:
If f: G-> H is a homomorphism then G/ker(f) is isomorphic to im(f)
The Attempt at a Solution
I understand that I need to show...
abstract algebra ...HELP, PLZ!
THIS IS THE PROBLEM: COMPUTE THE INDICATED QUANTITIES FOR THE GIVEN HOMOMORPHISM
KER (PHI) AND PHI(18) FOR PHI: Z -> Z10 (SUBCRIPT) SUCH THAT PHI(1)=6
Can anyone please help me to solve this problem? I don't even know what it's asking for? Don't know where...
I'm going to buy A First Course in Abstract Algebra by Fraleigh. I've looked at 6th and 7th ed. 6th doesn't have a section on homology groups, but 7th does. From what I found from other threads here, 4th also has homology groups, and 3rd is at least good on group actions. (I haven't got to group...
Homework Statement
Let R = { [ a + b*sqrt(m) c + d*sqrt(m) ] }
[ n(c - d*sqrt(m)) a - b*sqrt(m) ]
(Sorry if the matrix is unclear... I can't get it space nicely. r11 = a + b*sqrt(m) r12 = c + d*sqrt(m)
r21 = n(c -d*sqrt(m))...
One of my homework problems asks me to list the left coset (1,2,3)H where σ=(1,4,5)(2,3) and H=<σ>.
I know that you have to take the do the permutation of (1,2,3)(1,4,5)(2,3) but i am not sure how you can do that? I got (1,2,3)H={(1,2,3)(3)(1,2,4,5)} but i do not think that is right
The problem states:
Let R and S be nonzero rings. Show that R x S contains zero divisors.
I had to look up what a nonzero ring was. This means the ring contains at least one nonzero element.
R x S is the Cartesian Product so if we have two rings R and S
If r1 r2 belong to R and s1 s1...
Abstract algebra--> Let R be a ring and let M2(R) be the set of 2 x 2 matrices with
Homework Statement
Let R be a ring and let M2(R) be the set of 2 x 2 matrices with entries in R.
Define a function f by:
f(r) = (r 0) <----matrix
...(0 r)
for any r ∈ R
(a) Show that f is a...
Homework Statement
If G is a group with operation * and \alpha,\beta\in G, then \beta\ast\alpha\ast\beta^{-1} is called a conjugate of G. Compute the number of conjugates of each 3-cycle in S_{n} (n\geq3).
Homework Equations
The Attempt at a Solution
For any group S_{n} there...
abstract algebra question??
here is the problem from abstract algebra, anyone could help? Thanks a lot!
let G be a finite group. Show that in the disjoint cycle form of the right regular representation Tg(x)=xg of G each cycle has length | g |.
(Tg(x) means T sub g of x)
loofinf...
1. The problem statement:
Consider 3 positive integers, a, b, c. Let d_{1} = gcd(b,c) = 1. Prove that the greatest number dividing all three of a, b, c is gcd(d_{1},c)
3. My go at the proof and thoughts:
Well, I know that the common divisors of a and b are precisely the divisors of...
Homework Statement
We have a vector space (V, R, +, *) (R being Real numbers, sorry I couldn't get latex work..) with basis V = span( v1,v2). We also have bijection f: R² -> V, such as f(x,y) = x*v1+y*v2.
Assume you have inner-product ( . , . ): V x V -> R. ( you can use it abstractly and...
The question states prove,
If p is prime and p | a^n then p^n | a^n
I am pretty sure I have i just may need someone to help clean it up.
There are two relevant theorems i have for this.
the first says p is prime if and if p has the property that if p | ab then p | a or p | b
the...
I need to find all the cosets of the subgroup H={ [0], [4], [8] ,[12] } in the group Z_16 and find the index of [Z16 : H].
Help would be appreciated :)