Question

consider the multivariable polynomial ring q[x, y] and the ideal i = (x, y). prove that i cannot be a principal ideal. [hint: prove this by contradiction. suppose that there was some polynomial p(x, y) such that i = (p(x, y)). what is the contradiction?]

Answer 1

Answer:

One approach could be as follows:

• The elements of i all have the form xf+yg, where f and g are elements of q[x,y].  Therefore all elements of i have degree greater than 0.
• Suppose that i = (x, y) = ( p(x, y) ) for some polynomial p.
• Since x ∈ i, there is a polynomial f such that x = fp.
• Since y ∈ i, there is a polynomial g such that y = gp.
• deg(x) = deg(fp) = deg(f) + deg(p), so deg(p) = 0 or deg(p) = 1.
• But all elements of i have degree greater than 0, so deg(p) = 1.
• From x=fp, deduce that p = ax, for some a ∈ q.  Also, y=gp implies p = by, for some b ∈ q.  This is a contradiction.