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2h
comment Generalized Sophomore's dream. Question about originality
Your English is generally good. Your main mistake is that you use commas too much. (Наверно Вы русский? Они люююбят запятые. :))
2d
awarded  Nice Answer
Jun
30
comment The Frobenius Trace for an elliptic curve
See Ireland and Rosen's number theory book: Lemma 1 section 4 chapter 9 and Theorem 4 section 3 chapter 18.
Jun
30
comment For a group epimorphism $f : G \to H$ with kernel $K$, prove that $G \simeq K \rtimes H$. Why is $G \simeq K\times H$ if $G$ is abelian?
This is false. Let $G$ be cyclic of order $p^2$ for a prime $p$, $K$ be its subgroup of order $p$, and $H = G/K$, with the map $G \rightarrow H$ being reduction $g \mapsto gK$. The assertion would imply $G$ is isomorphic to a product of two groups of order $p$ and that is false.
Jun
27
comment Does an inseparable extension have a purely inseparable element?
Lipman wrote a short paper ("Balanced Field Extensions," Amer. Math. Monthly 73 (1966), 373-374) about the algebraic extensions that are separable extensions of inseparable extensions and at the end he gives a counterexample that is precisely this construction when $p = 2$ except he allows any field of characteristic $p$ in place of $\mathbf F_p$, which of course works in the above answer as well.
Jun
25
revised Help with proof that that affine plane curves in $\mathbb{C}^2$ are not compact
added 1 character in body
Jun
24
comment Alternate method to calculate an infinite string of numbers that's not $\pi$, and contains any string
What do you need this for? One answer mentions the Champernowne constant .12345..., which certainly works but has no more or less content than just listing all decimal strings of length 1, 2, 3,... in a specific order and can't really be "defined" in any way other than its decimal expansion, so it seems useless.
Jun
24
comment Example of two field extensions such that their tensor product is not a field
Your "map" is incorrect. It is just symbol-pushing and has no meaning. In fact your $K \otimes_k L$ is a field, namely $\mathbf Q(\sqrt{2},i)$. For an example try taking $L = K$.
Jun
24
comment Show that $\mathbb{Q}(\sqrt{5}+\sqrt[3]{2})=\mathbb{Q}(\sqrt{5},\sqrt[3]{2})$
There is a fairly general result in this direction, for fields of characteristic $0$. See the accepted answer at mathoverflow.net/questions/26832/….
Jun
24
comment Proof that algebraically closed fields of characteristic $p$ exist
@Stefan, the construction at the link is like Artin's argument but explicitly only needs one step, rather than the infinitely many steps that Artin's own method used (even though it turns out to be redundant after the first step).
Jun
22
comment Why is it that if a function is continuous, then $\lim_{x \to c} f(x) = f(c)$?
I suspect the OP is using the naive "definition" that continuity means you can draw the graph without lifting the pencil (or pen) off the paper.
Jun
19
comment Why do we use $\mathbb{R}$?
Ultimately there are systems of "numbers" going in different directions in which none are naturally contained in any other, such as the $p$-adic numbers for different primes $p$ or the finite fields $\mathbf Z/(p)$ for different primes $p$. (And finite fields are very useful in coding theory, so it's not as if these constructions are pure abstractions.) That you only conceive of number systems inside the complex numbers is because the kind of education in math that many people receive only operates within them. Not by a long shot is the path "up to the complex numbers" the only way you can go.
Jun
19
comment Favorite problems that lead to interesting diophantine equations?
The Mathematical Atlas (and links to anything on Dave Rusin's webpages) now seems to be dead.
Jun
17
answered Primitive $p^n$-th root of unity in $\bar{\mathbb{Q}}_p$.
Jun
16
comment A slightly stranger Hensel's Lemma
What's an example where this "stranger" Hensel's lemma is useful?
Jun
14
comment root of unity in a local field
Try Serre's "Local Fields." There is a section in it on cyclotomic extensions of $\mathbf Q_p$. Or better yet, just try working this stuff out for yourself. It is an instructive experience.
Jun
13
comment root of unity in a local field
Have you ever read a book on local fields?
Jun
13
comment root of unity in a local field
The prime-to-$p$ part of $n$ divides $q-1$, where $p$ is the residue field characteristic.
Jun
13
comment root of unity in a local field
Try $\mathbf Q_2$ and $n=2$. If you take $n$ to be relatively prime to $q$ then your guess is right.
Jun
11
comment Help with understanding a proof concerning traces of a Galois extension
The trace map can be defined for any finite extension of fields, not just Galois extensions. See math.uconn.edu/~kconrad/blurbs/galoistheory/tracenorm.pdf and math.uconn.edu/~kconrad/blurbs/galoistheory/tracenorm2.pdf (the second file discusses the link to the formula you use in the case of a Galois extension.