Answer by R.P. for Is every number field generated by a trinomial?
The answer of @wojowu leaves open the case whether there exist field extensions $K/\mathbb{Q}$ of degree $4$ that can't be generated by a root of a trinomial. Although the question has been answered...
View ArticleComment by R.P. on Copies of $\mathbb{R}$ in $\mathbb{C}$ using the axiom of...
Maybe good to add (as a little explanation) that for any $f$ that does not restrict to the identity on $\mathbb R$, we must necessarily have $f(\mathbb R)\neq\mathbb R$, because $\mathbb R$ has no...
View ArticleComment by R.P. on Tate/semisimplicity conjecture for resolution of nodal...
See also here: mathoverflow.net/questions/200129/…
View ArticleComment by R.P. on Is every number field generated by a trinomial?
So if I understand correctly, the same answer also proves that there does not exist $N$ such that every finite extension $K$ of $\mathbb{Q}$ can be generated by a polynomial containing at most $N$ terms?
View ArticleComment by R.P. on To which "lady-mathematician" is Weil referring?
@LSpice They do exist.
View ArticleComment by R.P. on To which "lady-mathematician" is Weil referring?
Somehow it doesn't exactly read as if Weil was particularly impressed by this female mathematician... :)
View ArticleComment by R.P. on Factorization of polynomials vanishing on quadrics:...
I think this should follow from Hilbert's Nullstellensatz. Since $g$ is irreducible, the ideal $I$ generated by $g$ is a prime ideal, therefore $I$ is a radical ideal, and it is a direct consequence of...
View ArticleComment by R.P. on Checking that $2$ is a primitive root modulo $p$ without...
@GHfromMO "I don't see a way to formalize it" doesn't imply "there does not exist a way to formalize it." I have seen many mathematicians dismiss a question or statement on the grounds that they can't...
View ArticleComment by R.P. on Extremely messy proofs
To me it looks like only slightly over four pages (pp. 474-478), and even then we should account for H.'s old-fashioned leisurely style, e.g. he treats the case n=2 as an illustration, and this is then...
View ArticleComment by R.P. on A Diophantine equation that is unsolved on a French forum
@FedorPetrov In the linked comment the poster seems to aver that $(5t)^2=(5s_2-21u)^2+56u^2$ has a sign error, and the last plus sign should be a minus sign.
View ArticleComment by R.P. on Can the Pythagorean theorem be proved using imaginary...
Sorry, but what does it mean that distance "is not a defined quantity but part of the model"? I feel this answer is not quite self-contained, even taking the comments into account.
View ArticleComment by R.P. on Rational independence of higher golden ratios
This looks like it's a piece of folklore that has been adapted into a homework problem.
View ArticleComment by R.P. on Average number of $\mathbb{F}_p$-points over twists of a...
This may be less deep than you think. For example Fact 1 can be simply explained by noting that every point on the projective line can be lifted either to two points on $E$ or to two points on its...
View ArticleComment by R.P. on Bounds for the difference in the number of ones in $M$ and...
So the maximum difference for $n=6$ is $16$ and the maximum for $n=7$ is $25$. Now I wonder if it will be $36$ for $n=8$... By the way do you have data for $n<6$ as well?
View ArticleComment by R.P. on Looking for review of delay differential equations...
Wouldn't a logarithmic substitution on the time domain transform this equation to a delay-differential equation of the more well-known type?
View ArticleComment by R.P. on In what respect are univalent foundations "better" than...
This whole turn towards homotopy type theory has me starting to think the other way: maybe putting mathematics on the foundations of set theory was a big mistake, and by fixing it in this way we are...
View ArticleComment by R.P. on Classification of Étale algebras without Galois theory and...
Maybe the much-fabled Lenstra lectures notes on Galois theory for schemes? (Theorem 1.11 seems to be what you want, except it is even a further generalization.)...
View ArticleComment by R.P. on Applications needing different constants of integration on...
I think this is a fair question, howbeit of a somewhat elementary nature. I am surprised to see three votes to close within the hour.
View ArticleComment by R.P. on The Hasse-Weil inequality
The phrase "points at infinity" is a bit ambiguous, since it depends on which projective embedding you are thinking of. If you're thinking of $\mathbb{P}^2$, then both curves intersect in the rational...
View ArticleComment by R.P. on What does keep you "doing what you do"?
Jerry Seinfeld offers the following career advice: find the torture you're comfortable with. We can't really eliminate struggle from our lives. In this world we will have trouble...
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