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Dec
10
comment Quadratic extensions - understanding
Yes, every finite extension is algebraic. If $L/K$ has degree $n$, then for any element $\alpha \in L - K$ the set $\{ 1, \alpha, \alpha^2, \ldots, \alpha^n \}$ must be linearly dependent, which shows that $\alpha$ satisfies a polynomial, i.e. $\alpha$ is algebraic.
Oct
14
comment Proving Logarithims
Forgive me for trying to read your professor's mind, but I imagine what she means is that you cannot start with the assumption that $A=B$, manipulate both sides simultaneously, and then arrive at a true statement. However, it is perfectly fine to take the log of both sides, $\log(A)$ and $\log(B)$, separately, and then observe that they are equal.
Sep
5
comment What does Linear mean in Linear Space (Vector Space)
A circle is a 1-dimensional space which is not flat. A (hollow) sphere is a 2-dimensional space which is not flat.
Jun
27
comment The field fixed by inertia group is the maximal unramified field
You can reduce to the finite case: Take an arbitary $\alpha$ in $K^I$ and consider the finite extension $F(\alpha)/F$.
Jun
16
comment Visualising finite fields
Yes, that's the paper. Gassert is currently at the University of Colorado at Boulder as a postdoc. Here's the full list of his arXiv papers: arxiv.org/find/math/1/au:+Gassert_T/0/1/0/all/0/1
Apr
1
comment $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$?
Have you tried computing the minimal polynomial for $\sqrt{2+\sqrt{2}}$?
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.