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1. Simple magmas?

There is a standard definition of a simple object in a finitely complete category with an initial object 0. See: http://ncatlab.org/nlab/show/simple+object for the general definition. A congruence on a magma M is an equivalence relation ~ on M such that a ~ b and c ~ d implies ac ~ bd. For any...
2. Inverse of a matrix

The Euclidean algorithm only works for relatively prime elements, so when he says that r(\lambda), q(\lambda) exists such that r(\lambda)(\lambda^3-8\lambda)+q(\lambda)(\lambda^2+1)=1 he uses that they are relatively prime.

Yes.
4. Coninvolution of a Matrix

It is unclear what you are asking for. Are you asking 1) How to prove your lemma, 2) for examples of coinvolutory matrices, or 3) for general litterature on coinvolutory matrices? The lemma is fairly straightforward to prove by calculating \left(\left(\overline{A}\right)^{-1}A\right)^{-1} and...
5. Non-singularity of A^T*A

Is X a square matrix? If so use det(X^TX) = det(X)^2=0 If not X is not square, but is real, then QR decomposition should reduce the problem to that of square matrices (something simpler may suffice, but this is the simplest approach I can think of right now).
6. Group structure definition

Could you perhaps tell us the actual example. Structure can mean slightly different things to different people in different contexts. It can mean high level concepts such as whether the group is Abelian (as Benn indicates a^2 = 1 implies that it must be), but in my experience this is not what is...
7. Modern Algebra: Permutations and Cycles

Usually what is meant is for you to show this for arbitrary permutations. You need to show that \theta and \theta^{-1} have the same cycle structure no matter what permutation \theta is. Some ideas for this may be gained by considering specific examples like (1 2 3) (1 2)(3 4) etc. and seeing...
8. Free Group with defining equations

I can obviously not be sure what example your professor had in mind, but I recall an example that may be of a similar spirit (this is from Dummit-Foote if I recall correctly). \langle x,y|x^n=y^2=e,\quad xy=yx^2\rangle Here one may guess that this group has order 2n, but x = xy^2 = yx^2y...
9. Does doubling the sum of prime factors always lead to 16?

16 is the only ending number. RamaWolf showed that for small numbers we always end up at 16 so I will just show that we always get to a small number (we can take small to mean <485, but if we are not lazy or didn't have RamaWolf's calculations we can manually reduce to much lower numbers) and...
10. Difficult linear algebra problem

This doesn't sound like homework so I will assume it's not (and therefore not feel bad about providing a "solution"). I'm not sure exactly what you mean by a shifting matrix. The only definition I know of is matrices which are 0 everywhere except on precisely one diagonal either below or...
11. Can't get my head into vector spaces and subspaces

The problem you gave is hard because it is false. V = \{(x,y,z) \in \mathbb{R}^3 | xy = 0 \} is not a subspace of \mathbb{R}^3, in particular if (x,y,z) and (x',y',z') are in V you cannot be certain that (x+x',y+y',z+z') is in V. If you can, you should try to see why this is (that is come...
12. R^2 a field?

EDIT: Completely ignore this. Didn't think it through. For an example of why it's false see Office Shredder's reply. Yes if R is a field, then R^2 is a field (clearly commutative, and (a,b) has inverse (1/a,1/b) ).
13. Definition of normal extension

Yes the family can be arbitrarily large.
14. Definition of normal extension

What is the splitting field of all polynomials with coefficients in Q? If you can show that this is E, then you have shown that E/Q is a normal extension.
15. Division algorithm in A[x] (A NOT a field!)

I'm assuming you mean to require deg r < deg f (not the other way around). No division of coefficients is required as long as f is monic (you don't need that A is local either). If f has degree larger than v you can just let r=f, q=0. Thus you can assume deg(f) \leq deg(v). Let (v,f) be a...
16. Is randomness provable?

If that was the case then again what does it mean for a sequence to be random? Is (2,3,4) random? If what is meant is to determine whether the sequence was generated randomly, then of course you can't determine that. If I tell you the sequence 2,4,6,... of even positive integers you can't tell...
17. Is randomness provable?

What does this mean? I don't see how this makes any sense. It's not really meaningful to ask whether the integers are random. That depends on what you mean by a random integer. An integer is a particular one. You can't have an integer that is 50% 1 and 50% 3. However what you can have is a...
18. Can 1+1 = 0?

If 1=0 in a field, then for every element x of the field we have: x = 1x = 0x = 0 so the only field in which we can have 1=0 is the trivial field {0}. Thus we can assume 1\not=0 and our theory will still cover all non-trivial fields. This is why most books don't care about the case 1=0. If you...
19. Background needed to understand number theory research papers.

Yes a lot of them will. And those that don't will have the necessary background to pick it up easily. That is not really something you need to answer now. You do probably not know what goes into studying the cohomology of number fields, or even what exactly algebraic number theory and analytic...
20. Background needed to understand number theory research papers.

(I'm not myself capable of doing research in number theory yet, but I have some knowledge of what it takes as I'm considering specializing in it. Take my advice with a grain of salt as I may not really have the authority to speak on this matter.) It depends on the kind of research. Number...
21. Pairs of twin primes

1481, 1483, 1487, 1489 is the first counterexample. (1491 = 41 + 90 * 16) However what is true is that they are all of the form: 30k + 11, 13, 17, 19. This can easily be proven by supposing we have primes n+11,n+13,n+17,n+19 (with n non-negative). n must be even because otherwise n+11 is...
22. 2 number theory problems

0,1,2,...,s are s+1 numbers so there are s+1 terms, and thus C_s contains 3^(s+1) elements. The maximal value of an element of C_s is: N=3^0 + 3^1 + \cdots + 3^s = \frac{3^{s+1}-1}{3-1} = \frac{3^{s-1}-1}{2} and the minimal is -N. 2N+1 = 3^(s+1).
23. 2 number theory problems

Second problem: I assume you know the identity (otherwise you can easily show it): L_n = F_{n-1} + F_{n+1} Then your identity is equivalent to: F_{2n} = F_nF_{n+1}+F_nF_{n-1} Now let us strengthen to induction hypothesis to also say: F_{2n+1} = F_{n+2}F_{n+1}-F_nF_{n-1} Then you can...
24. 2 number theory problems

First problem: The way I proved it was to prove for a fixed s, every integer in n with absolute value less than or equal to 3^0 + 3^1 + \cdots + 3^s can be expressed uniquely in the form: \sum_{j=0}^s c_j 3^j with c_j \in \{-1,0,1\} (I don't require that c_s is non-zero, since if it is we...
25. Method for rotating data points in 3 dimensions

So these are voxels? Then it would be misleading to say that they represents points of certain objects in a game. If you truly had an array representing points in a game your array would probably look more like: int [10,3] currentLocs; for 10 objects in 3-dimensional space. If this is really...
26. Fermit's theorem

I'm assuming you don't know or don't want to use Euler's theorem. Note 4010 = 2*5*401. Can you find integers a,b,c such that \begin{align*} 5^{2005} &\equiv a \pmod 2 \\ 5^{2005} &\equiv b \pmod 5 \\ 5^{2005} &\equiv c \pmod {401} \end{align*} ? (perhaps using Fermat's little...
27. Most challenging problem from I. N. Herstein's Algebra Book

Suppose x is central and n is a positive integer. Then for any other element y we have xy=yx and therefore: \begin{align*} (nx)y&=(x+x+\cdots+x)y \\ &= xy+xy+\cdots+xy \\ &= yx + yx +\cdots + yx \\ &= y(x+x+\cdots+x) \\ &= y(nx)\end{align*} which shows that nx is also central. The same...
28. Expansion of Fermat's Little Theorem

BTW you are probably forgetting some requirements, because as stated your identities are false. Take for instance a=4, b=6, then you get: \phi(a)=\phi(b)=2 a^{\phi(a)}+b^{\phi(b)}=4^2+6^2=52 \equiv 4 \pmod {ab=24} Perhaps you meant for a and b to be relatively prime. And in the first you can...
29. Expansion of Fermat's Little Theorem

Can you show: p^{q-1}+q^{p-1}\equiv 1 \pmod{p} ? If so can you do the same mod q and combine these results to get it mod pq?
30. Question on the set of zero-divisors of a ring

I didn't think properly. I for some reason assumed the ring was commutative so please ignore my previous post.