Subring of polynomial ring containing two polynomials of relatively prime degree

Question

Let $R$ be an integral domain , $S$ be a subring of $R[x_1,..., x_n]$ .

If $S$ contains two polynomials of relatively prime degree, then how to show that there exists an integer $m>1$ such that $S$ contains a polynomial of degree $l$ for every $l>m$ ?

I can easily show the claim for $n=1$, but having difficulty in showing for higher $n$. Please help.

How did you do it for $n=1$? – Eric Wofsey Mar 06 '19 at 22:58 — Eric Wofsey, Mar 06 '19 at 22:58

score 3 · Accepted Answer · answered Mar 06 '19 at 22:59

3

Hint: Prove that if a subset $A\subset\Bbb{N}$ is closed under taking sums, and if $A$ contains two relatively prime elements, then there exists an integer $m>1$ such that $A$ contains all natural numbers $l>m$.

answered Mar 06 '19 at 22:59

Servaes

67,306
8
82
171

Relevant: The Frobenius coin problem. – Nothing special Mar 02 '25 at 14:01

Subring of polynomial ring containing two polynomials of relatively prime degree

1 Answers1

Linked