# Problem 10437 from the American Mathematical Monthly...

(I submitted this problem in the summer of 1993, while I was a grad student at U. C. Berkeley and enjoying a summer job at AT&T. The problem eventually appeared in the February 1995 issue (Volume 102, #2, page 170) of the ``Monthly.'' No correct solutions were received, so my solution was finally published (in the same journal) in 1997.)

Let R be a ring (whose multiplication need not be commutative or associative) without zero divisors. Let x_1,...,x_n be algebraically independent indeterminates over R which commute and associate amongst themselves and commute with the elements of R. Also assume the associative law for products of one element of R, an x_i, and an x_j. Prove the following:

(a) If f is a homogeneous polynomial in R[x_1,...,x_n], then every divisor of f is homogeneous.

(b) If a_1,...,a_n are nonzero elements of R and d_1,...,d_n are nonnegative integers with gcd(d_1,...,d_n)=1, then the polynomial

a_1x^{d_1}_1 + ... + a_n x^{d_n}_n

is irreducible in R[x_1,...,x_n], i.e., every factorization has at most one nonconstant factor.