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Apr
1
comment $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$?
Have you tried computing the minimal polynomial for $\sqrt{2+\sqrt{2}}$?
Mar
31
comment A question regarding Kummer
I don't understand your question, and I honestly don't know the history of algebraic number theory, so I'm not sure whether ideal class groups originated with Kummer. But Kummer's example of failure of unique factorization is frequently the example used to motivate ideal class groups.
Feb
11
comment Finitely generated algebra as a module over a subalgebra
Two polynomials (in $f_3$) are equal if and only if all of their coefficients are equal, so this is immediate.
Feb
10
comment The mod $p$ Galois representation of the Frey curve is unramified away from $2, p$
This is an excellent answer.
Feb
9
comment The mod $p$ Galois representation of the Frey curve is unramified away from $2, p$
The one-line "intuitive" answer to your question is that the primes dividing the discriminant are those for which the curve has bad reduction, i.e. reducing modulo those primes yields a singular curve over a finite field, so we expect bad things to happen at such primes. Where the curve is smooth, we expect everything to be nicely behaved.
Oct
6
comment Finding the area of an ellipse using linear algebra
I guess it's impractical for me to hope to police this entire site, so never mind. I have external ways to deal with this situation.
Oct
6
comment Finding the area of an ellipse using linear algebra
Yes, I'm ok with hints, and I like yours. What I'm concerned about are full solutions.
Oct
6
comment Finding the area of an ellipse using linear algebra
Also, the user has already asked another question which was completely answered for him/her. Clarifications are one thing (and easily obtained my emailing me). Solutions are a different matter.
Oct
6
comment Finding the area of an ellipse using linear algebra
@dleggas While I take your point, this particular assignment is technically a take-home quiz. Certainly you agree that questions from take-home quizzes and exams should not be answered on this website?
Oct
6
comment Finding the area of an ellipse using linear algebra
This is an assignment which I have given as homework. Please do not answer this. My students should come to me for help.
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