On subalgebras of n×nn\times n matrices not satisfying identities of degree 2n−22n-2

The Amitsur-Levitski theorem asserts that $M_n(F)$ satisfies a polynomial identity of degree $2n$. (Here, $F$ is a field and $M_n(F)$ is the algebra of $n \times n$ matrices over $F$). It is easy to give examples of subalgebras of $M_n(F)$ that do satisfy an identity of lower degree and subalgebras of $M_n(F)$ that satisfy no polynomial identity of degree $\le 2n-2$. Our aim in this paper is to give a full classification of the subalgebras of $n \times n$ matrices that satisfy no nonzero polynomial of degree less than $2n$.Comment: 11 page

Factorization of quadratic polynomials in the ring of formal power series over Z\Z

We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring $Z[[x]]$ of formal power series with integer coefficients. For $n,m\ge 1$ and $p$ prime, we show that $p^n+p^m\beta x+\alpha x^2$ is reducible in $Z[[x]]$ if and only if it is reducible in $Z_p[x]$, the ring of polynomials over the $p$-adic integers.Comment: 15 page

Factoring polynomials in the ring of formal power series over Z

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility as power series. Moreover, if a polynomial is reducible over Z[[x]], we provide an explicit factorization algorithm. For polynomials whose constant term is a prime power, our study leads to the discussion of p-adic integers.Comment: 10 pages, submitte