I had never really thought about this. All I know is that it will be true when M is a prime number, or if N is a number with a square-free prime factorization ( use eisenstein's ). I don't if there are easier conditions
edit: also, I think it will be true if M is not a power of 2. Over Q, you...
There isn't anything wrong with saying it like that. That's the idea: if p is irreducible over F, then p is a minimal polynomial ( once you scale it to be monic ), of some x which is a root in the splitting field. If x is also a root of Dp, then the minimal polynomial of x has to divide Dp. As...
It is false, try R = Z and I = ( 3 ), then R/I is not a subring of Z.
To prove this, remember that a subring must also be a subgroup of the additive abelian group that makes up the ring R.
This is because every field F of characteristic zero is a perfect field. This means that every irreducible polynomial over F is separable ( splits in the splitting field with distinct linear factors ).
A polynomial p(x) has multiple roots in the splitting field, iff the polynomials p(x)...
A lie group isomorphism f between lie groups G and H will have full rank, so that the corresponding map between lie algebras df is an isomorphism. However, the converse is false, two lie groups can have the same lie algebra but be non isomorphic lie groups.
Groups that are locally diffeomorphic...
A homomorphism of Z_3 into S_3 must be an injective homomorphism, as Z_3 does not contain any non-trivial subgroups [ suppose f is a homomorphism from Z_3 into S_3, then |im( f )| = | Z_3 / ker(f ) | i.e. the image of f must divide |Z_3| = 3. So it is either injective, or trivial ].
Now, in...
This is when your manifold is not already embedded in another topological space ( for example, R^N ). If you notice, the way WBN defined "locally euclidean", he said gave a condition that the coordinate maps must be homeomorphisms with respect to a prescribed topology on X.
The most general...
Actually, what samalkhaiat is said is true. Any connected Lie group is generated by any neighbourhood around the identity ( that is, take an open neighbourhood U of the identity and consider all possible products and inverses ). This is because the subgroup generated by an open neighbourhood U...
( a related note is the word problem for groups: given a finite set of generators and a presentation, one cannot algorithmically decide whether or not two elements "written down" represent the same element in the group [ sorry if I butchered the statement ]. Anyway, the point is, deciding...
I'm not entirely sure what you mean by this question.
In "reality" a group will not have "repeated elements" (whatever this means ), but there are multiple ways to write down the same element in a group. For example, in a dihedral group,
sr^-1 and rs are two ways to write down the same...
https://www.amazon.com/dp/0120502577/?tag=pfamazon01-20
this one? It's on a higher level than those 3 you've listed. Baby Rudin hardly talks about measure