56,074 reputation
473158
bio website users.utu.fi/lahtonen/…
location Rusko, Finland
age 51
visits member for 3 years, 6 months
seen 1 hour ago

General non-sense:

Mostly I teach here. I want to encourage beginning students to think for themselves, so I use a lot of hints and comments. More advanced questions I often just answer.

Relevant personal history:

PhD from Notre Dame in '90.

Drifted from representation theory of algebraic groups to applications of algebra into telecommunications, mostly coding theory, and lately mostly teaching at college level.

3 graduate students with awarded PhDs

I have mostly worked at our local University at Turku, Finland. At one point I tried working for Nokia Research Center. It was ok, but an old dog didn't learn all the tricks, and then they downsized, so I returned to the Uni as a tenured lecturer.


7h
comment A cubic equation: $u^3−2u^2−2v^3−20v^2+16v=0$
An algorithm for finding the minimal Weierstrass form is described here. See this MO question. I was doing some trial-and-error fumbling, but couldn't get to the minimal 2-adic model :-( May be PARI/GAP has this implemented?
10h
comment A cubic equation: $u^3−2u^2−2v^3−20v^2+16v=0$
Related.
10h
comment A cubic equation: $u^3−2u^2−2v^3−20v^2+16v=0$
As a way of confirmation. Both curves have $j$-invariant equal to $$j=-953312/9075.$$ Thus we can get one from the other. Hang in there!!!
10h
comment A cubic equation: $u^3−2u^2−2v^3−20v^2+16v=0$
If you substitute $x=16X$ and $y=64yY$ in your equation, you get $$Y^2=X^3-10X^2-8X+528.$$ Both this equation and the equation $Y^2=X^3-X^2-41X-441$ have discriminants $-2^{13}\cdot3\cdot5^2\cdot11^2$, which is too much of a coincidence. Further minimizing may thus be possible at $p=2$. Trying to recall how that is done :-)
12h
revised Uniqueness of an embedding theorem for Real differential fields
edited tags
1d
comment Proving that $T$:$(x_1,…,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},…,\frac {x_n+x_1}{2})$ leads to nonintegral components
But is it not trivial that if a linear transformation $T$ maps all the integer vectors to integer vectors, then the matrix of $T$ (w.r.t. the natural basis) consists of integers as its columns are the images of the natural basis. Is it because you also want to use the argument in that $T$-invariant subspace of codimension one, where the natural basis is not available?
1d
comment $R$ is the ring $\mathbb{Z}[\sqrt{-k}]$. In $R$, if $3\mid (a+b\sqrt{-k})$, then $3\mid a$ and $3\mid b$ in $\mathbb{Z}$
Have you tried using the definition of divisibility? If $3\mid a+b\sqrt{-k}$, then there exists an element of $R$ such that ...
1d
comment Proving that $T$:$(x_1,…,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},…,\frac {x_n+x_1}{2})$ leads to nonintegral components
Could you elaborate on your argument proving that $\det T$ has to be an integer? Exactly what are you assuming about $T$ at that point? That it maps any integer vector with pairwise distinct entries to an integer vector not constrained in that way? Or what?
1d
awarded  Constituent
2d
comment Bounds for general character sums over finite fields
Rosen does give Bombieri's version of the Schmidt-Stepanov method of proving the Riemann hypothesis for function fields. (If you are a coding theoretically minded, then a similar account is in Stichtenoth's book). Combining this with what you learn about L-functions from the other listed sources will go a long way. For example, the Shanbhag-Kumar-Helleseth extensions I refer to in my answer become "straightforward" exercises.
2d
comment Bounds for general character sums over finite fields
Schmidt's book is very useful (our library copy is also mysteriously misplaced). Lidl & Niederreiter don't go into details here even though they explain the use of L-functions IMO a bit better. I benefitted from studying Michael Rosen's book in the sense that there is a more number theoretical account there. Unfortunately he doesn't go into details on the character sums. But you will recognize that those mysterious subgroups of the multiplicative group of $\Bbb{F}_q(x)$ are related to ray classes.
2d
revised Bounds for general character sums over finite fields
edited tags
2d
answered Bounds for general character sums over finite fields
2d
reviewed Approve NTRU cryptosystem
2d
revised How prove this diophantine equation $x^2-y^2\equiv a\pmod p$ have only $p-1$roots
edited tags
Dec
18
comment Galois finite extension
No problem, @Adam. I had skipped over a part, which I assumed to be common knowledge, but which in a way is at the heart of the question.
Dec
18
revised Galois finite extension
added 304 characters in body
Dec
18
comment Galois finite extension
If you don't want to assume that $K$ is a subfield of $\Bbb{C}$ in the beginning, then a more accurate argument would be that we pick an embedding $f:K\to\Bbb{C}$, and then we get $n$ different, i.e. all, embeddings with the recipe $f\circ\sigma$ with $\sigma$ ranging over the group of automorphisms of $K$. Then all these embeddings share the image with $f$.
Dec
18
comment Galois finite extension
@Adam. Agree with your first point in that I tacitly picked one embedding to turn $K$ into a subfield of $\Bbb{C}$. But why would I need the automorphisms of $K$ to act on all of $\Bbb{C}$? They never need to be defined outside $K$, or if you prefer $\sigma_1(K)$.
Dec
18
comment Galois finite extension
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