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Jun
26
comment Can a ring without a unit element have a subring with a unit element?
www-math.mit.edu/~poonen/papers/ring.pdf
Apr
8
comment if B is a maximal linearly independent set in V then B is a basis for V
It is a complete proof. Think about it some more.
Apr
8
comment if B is a maximal linearly independent set in V then B is a basis for V
Yes. I don't understand your line of questioning. Are you unfamiliar with proofs by contradiction?
Apr
7
comment if B is a maximal linearly independent set in V then B is a basis for V
That is the contradiction. ($B \cup \{w\}$ is linearly independent by assumption.)
Nov
29
comment Abstract Algebra: Field extensions
Right. The norm will be fixed since the composition of two automorphisms $\sigma_i$ and $\sigma_j$ is another automorphism $\sigma_k$.
Nov
25
comment Important topics to cover in an intro course to algebraic number theory?
I can't upvote this answer enough.
Oct
3
comment Properties of Linear Transformations?
Very often, the motivation for studying a map $T:A \to B$ between algebraic objects is a scenario where we understand $B$ very well, and we wish to understand $A$ better. In order to be able to transfer information from $B$ back to $A$, we need our map $T$ to preserve the relevant structures.
Oct
3
comment Properties of Linear Transformations?
Oh, and $\mathbf{R}^n$ is an example of a vector space.
Oct
3
comment Properties of Linear Transformations?
they are somehow less useful to study. If you study more mathematics, you will encounter many more examples of this sort of thing, and it will make more sense then.
Oct
3
comment Properties of Linear Transformations?
Jared's answer is focusing on the intuition. Mathematicians study many different types of objects; a vector space is an example of what might be called an algebraic object. A vector space is composed of things called vectors, and there are two operations you can carry out with these vectors: you can add them together, or you can scale them. So if we want to study maps (that is, functions) between two vector spaces, the best ones to consider will be the maps which respect those two operations. Thus, we get the definition of a linear transformation. Other maps between vector spaces exist, but
Oct
2
comment Why learning modern algebraic geometry is so complicated?
I feel compelled to link to this post on MO regarding the usefulness of studying EGA: mathoverflow.net/questions/3041/…
Jun
28
comment Are $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ and $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ isomorphic as $\mathbb{R}$-vector spaces?
$\mathbb{C} \otimes _\mathbb{C} \mathbb{C} \simeq \mathbb{C}$
Jun
16
comment A subset of a field that is a subfield
This is definitely what the problem-writer had in mind.
Jun
11
comment '$R$-rational points,' where $R$ is an arbitrary ring
Yes, but that edit button is so far away...
Jun
11
comment '$R$-rational points,' where $R$ is an arbitrary ring
We seem to have posted at the same time, but I like your answer better =D
Jan
17
comment Congruence modulo's in a polynomial field being a field?
Do you know some ring theory? For a ring $R$ and an ideal $I$, $R/I$ is a field if and only if $I$ is a maximal ideal. So you could show that the maximal ideals in $F[x]$ are precisely those generated by irreducible polynomials.
Aug
7
comment Real world applications of prime numbers?
My comment (over a year ago...) about computational power was tongue-in-cheek. =D
Aug
1
comment Are there any synonyms of “pair of pants” in topology?
I was amused to find that the wikipedia page for "Trousers" does, in fact, have your wikipedia page as a disambiguation.
May
20
comment Confused about Wikipedia page on differential forms
Yes, that's correct.
Aug
26
comment Root of Unity Product
Well, I feel dumb. Thanks, anon!