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1d
awarded  Self-Learner
1d
accepted Easiest way to distinguish $E_8 \oplus E_8$ from $E_{16}$
1d
answered Easiest way to distinguish $E_8 \oplus E_8$ from $E_{16}$
Jul
2
awarded  Curious
Jul
1
comment Units of $\mathbf Z[X,Y]/(P(X,Y))$
Note that this is true only for reduced algebras. The "standard" counterexample is: $\mathbf Z[X,Y]/(X^2)$.
Jun
25
comment Easiest way to distinguish $E_8 \oplus E_8$ from $E_{16}$
@few_reps: is there an analogous command in pari/gp? (I know that there is an online Magma calculator, but for various reasons, it would be more convenient for me to do it in pari/gp).
Jun
24
comment integral equivalence classes of quadratic forms
Isn't the example of disagreement in dimension 16 due to Witt instead (and used by Milnor later for is counteraxample)?
Jun
24
comment Easiest way to distinguish $E_8 \oplus E_8$ from $E_{16}$
All this seems like a "rather complicated task". I mean: I am in front of a $16 \times 16$ matrix, with not so small integers. Pari/gp can list all 480 minimal length vectors, but analyzing this data will be tedious. I would have hoped for a trick.
Jun
24
asked Easiest way to distinguish $E_8 \oplus E_8$ from $E_{16}$
Jun
21
awarded  Yearling
May
27
comment Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$
Actually the reference by Ash--Brakenhoff--Zarrabi answers exactly my question.
May
27
comment Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$
In case it may interess someone: the proportion 60% found is conjectured by Lenstra to be $\frac 6 {\pi^2}$ (cf. the paper of Ash--Brakenhoff--Zarrabi mentionned in David Speyer answer).
May
27
accepted Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$
May
24
comment Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$
Thanks for the reference. Note that, however, numerical experiences seem to show that squarefree discriminants explain only half of the proportion obtained (and Stickelberger's relation does not help much).
May
23
awarded  Nice Question
May
23
revised Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$
added 67 characters in body
May
23
asked Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$
Apr
25
accepted Units of $\mathbf Z[X,Y]/(P(X,Y))$
Apr
23
asked Units of $\mathbf Z[X,Y]/(P(X,Y))$
Oct
24
comment Bass' stable range of $\mathbf Z[X]$
The answer is given in a paper of Grunewald, Mennicke and Vaserstein (On the groups $SL_2(\mathbf Z[x])$ and $SL_2(k[x,y])$). Israel J. Math. 86 (1994), no. 1-3, 157–193). One example of unimodular row that is not reducible is the following $(21+ 4x, 12, x^2 + 20)$.