Prove that the ideal of $\mathbb{R}[x]$ generated by $x^3-x^2+x-1$ and $x^4+3x^2+2$ is a prime ideal

Question

Prove that the ideal of $\mathbb{R}[x]$ generated by $f(x)=x^3-x^2+x-1$ and $g(x)=x^4+3x^2+2$ is a prime ideal.

Also, prove that the ideal generated by $r(x)=x^3-x^2-x+1$ and $s(x)=x^4+x^2-2$

Call the ideal generated by these two polynomials $P$. For $P$ to be prime we need that if $a(x)b(x) \in P$ then $a(x) \in P$ or $b(x) \in P$.

So i didn't make too much progress on this yet, hah, as of right now i'm trying to figure out a mistake I must have made in my algebra somewhere finding the roots of $g(x)$... So the roots of $g(x)$ are $(+/-)i,(+/-)\sqrt{2}$ but when i multiply the corresponding factors $(x-i)(x+i)(x+i\sqrt{2})(x-i\sqrt{2}$) I get $(x^2+1)(x^2+2)=x^4+4x^2+2$... so I don't know why the three keeps turning into a four.. Man i think i need a break.

Anyway if anyone can help me resolve the smaller problem which I have dug myself and the actual problem I asked that would be very helpful. Thank you!

You can note that $f(x)=x^2(x-1)+x-1=(x-1)(x^2+1)$ and $t^2+3t+2=(t+2)(t+1)$. — egreg, May 11 '19 at 15:00

score 3 · Accepted Answer · answered May 11 '19 at 14:46

3

Hint:

Over any field, the ideal generated by two polynomials is the ideal generated by their g.c.d. So use Euclidean algorithm to find the g.c.d. of these polynomials.

answered May 11 '19 at 14:46

Bernard

175,478

And when I find the $gcd$ what then? If the ideal is prime, will it the generator be an irreducible or osmething? – May 11 '19 at 14:51
Yes, you'll find an irreducible g.c.d. – Bernard May 11 '19 at 15:02

score 1 · Answer 2 · edited May 11 '19 at 15:03

Well, since ${\Bbb R}[x]$ is a principal ideal domain, the ideal generated by the polynomials $f,g$ is principal. This ideal has the monic generator $x^2+1$ which is irreducible in ${\Bbb R}[x]$.

Use polynomial division as below.

The ideal generated by $r,s$ is principal and has the monic generator $x^2-1$ which is reducible in ${\Bbb R}[x]$.

Polynomial division gives $r = (x+1)\cdot s + 3(x^2-1)$. So the ideal generated by $r,s$ contains $x^2-1$ which is the desired principal generator.

You can check the correctness by using Singular:

> ring r = 0, (x), lp;
> ideal i = x3-x2+x-1, x4+3x2+2;
> std(i);
_[1] x2+1
> ideal j = x3-x2-x+1, x4+x2-2;
> std(j);
_[1] = x2-1

Prove that the ideal of $\mathbb{R}[x]$ generated by $x^3-x^2+x-1$ and $x^4+3x^2+2$ is a prime ideal

2 Answers2