Abstract Definition and 506 Threads

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 For every ϕ in Aut(G), ϕ(Z(G))= Z(G). Homework Equations Z(G):={g in G| gh=hg for all h in G} The Attempt at a Solution I haven't made too much progress on this one. I know that if I let g be an element of Z(G) that I need to prove that For every ϕ(g) is also...

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...

What would be the best way to show that if F is an infinite field and f(x) is a polynomial in F[x] and f(a)=0 for an infinite number of elements a of F, that f(x) must be the zero polynomial? It kind of just makes logical sense to me, so I can't think of a way to actually show this. please help

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...

Hi All, Does anyone know of a way to make plots in mathematica in terms of variables? For example, suppose you had a function sin(a*x), and you wanted to plot it but did not want to set a to a specific value, the purpose being to have the graph report multiples of a, not specific numbers...

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...

I was wondering if there are any applications of abstract algebra to engineering and where I can go to learn about them?

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 assume there is no single right philosophy-of-physics (p-o-p) doctrine. No pretended philosophical "high ground". PoP is an academic subject--there are conferences and workshops. There is a fat two-volume Handbook of Phil. of Phys. published by North Holland Press. Courses are taught, seminars...

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...

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...

It was suggested that this belongs in philosophy so I have moved as I was enjoying the thread. Anyone have any views? ...Is the universe an abstract concept. Is time an abstract concept? Originally Posted by TheAlkemist View Post Yes and yes. An abstract concept = A mental...

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

An inner product space is often simply described as a vector space with the addition of an inner product, but when it comes to the formal definition, the basefield seems to always be restricted to the fields of real and complex numbers. The Wikipedia article on inner product spaces remarks that...

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...

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...

Homework Statement Determine whether the given function is one to one and whether it is onto. If the function is both one to one and onto, find the inverse of the function. f:R^{2}\rightarrowR^{2}, f(x,y)=(x+y, y) . Homework Equations The Attempt at a Solution I know one to one...

I am a junior in college right now, and after finishing the Calculus sequence and having my first semester of Analysis, I am now taking Abstract Algebra. I did alright in Analysis but not as good as I had hoped to do. My biggest problems are that, unlike Calculus, which for the most part I...

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...

Homework Statement Show that if a | c and b | c, and (a, b) = d, then ab | cd.Homework Equations The Attempt at a Solution Abstract divisibility. We have c=am, c=bn, and d=an+bm.

Homework Statement A positive integer a is called a square if a=n^2 for some n in Z. Show that the integer a>1 is a square iff every exponent in its prime factorization is even. Homework Equations The Attempt at a Solution Well, I know a=p1^a1p2^a2...pn^a^n is the definition of...

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...

Homework Statement Prove: (A-B)\cup(B-A)=(A\cupB)-(A\capB) Homework Equations The Attempt at a Solution We need to show (A-B)\cup(B-A)\subseteq(A\cupB)-(A\capB) and (A\cupB)-(A\capB)\supseteq(A-B)\cup(B-A). We begin by showing the first: Let x\in(A-B)\cup(B-A). By...

Homework Statement Find a statement for , S, equibalten to ~(P v Q) and show that it is logically equivalent by construction the truth table for "S if and only if ~(P v Q)" and showing that this statement form is a tautology. Homework Equations The Attempt at a Solution My...

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 :)

Homework Statement Let R be any ring and f:Z→R a homomorphism. a)Show that f is completely determined by the single value f(1) b)Determine all possible homomorphisms f in the case when R = Z. Homework Equations The Attempt at a Solution This question has me totally confused...

1. Suppose that H and K are distinct subgroups of G of index 2. Prove that H intersect K is a normal subgroup of G of index 4 and that G/(H intersect K) is not cyclic. 2. Homework Equations - the back of my book says to use the Second Isomorphism Theorem for the first part which is... If K...

Let R be a commutative ring. Show that the characteristic or R[x] is the same as the characteristic of R. I'm really not sure where to start on this at all. I'm not sure what is ment by R.

Let a, b be integers a,b>0 show that if a^3 | b^2 then a|b (Consider the prime factorization of a and b) I've tried setting up generic prime factorization of a and b but then don't get any where, I'm not very strong at this subject. Any kind of hints / where to start would help a lot...

How many groups are there of order 7. I just need to know this to continue with an assignment. Thanks.

Let G be a group. Show (xy)^{-1} = x^{-1}y^{-1} for all x, g \in G if and only if G is abelian. Homework Equations The Attempt at a Solution

Homework Statement Let f(x)=x5-x2-1 \in C and x1,...,x5 are the roots of f over C. Find the value of the symmetric function: (2x1-x14).(2x2-x24)...(2x5-x54) Homework Equations I think, that I have to use the Viete's formulas and Newton's Binomial Theorem. The Attempt at a...