# Find a generators for the following ideal in R[X]

$$I =\langle x^5+x^3+x^2-1, x^5+x^3+x^2 \rangle$$

I am struggling with this exercise as I have been given definition for ideals but no worked examples. I have also struggled to find any examples online that are clear to me. If possible could i have the method to solving such a question explained to me and what the question is asking for?

• Hint: $x^5+x^3+x^2- (x^5+x^3+x^2-1)=1$ – Arnaud Mortier Feb 20 '18 at 16:31

## 2 Answers

As the polynomial ring over a field is a P.I.D., the ideal generated by a family of elements is generated by a g.c.d. of these elements.

You can easily find $\gcd(x^5+x^3+x^2-1, x^5+x^3+x^2)$.

• If i am not mistaken the gcd=1. Does that mean the generator of I is 1 ? – Tim Jones Feb 20 '18 at 21:18
• Yes. These two polynomials are coprime, so they generate the whole ring of polynomials. – Bernard Feb 20 '18 at 21:23
• Is this the only one generator? if there are more how do i find them? – Tim Jones Feb 21 '18 at 12:26
• Any unit in $R$ will be a generator. Working in $\mathbf Z$, this means, you might as well take $-1$. – Bernard Feb 21 '18 at 12:35

Ideals have several properties:

1. An ideal is an additive subgroup of the ring. Therefore if an ideal contains $P$ and $Q$ it will contain $P+Q$ as well as $P-Q$. With $P=x^5+x^3+x^2$ and $Q=x^5+x^3+x^2-1$ you have $P-Q=1$. Therefore $$1\in I$$

2. An ideal is closed under multiplication by elements of the ring. This means that if $P\in R[X]$ and $Q\in I$ then $PQ\in I$. Applying this to any $P$ and $Q=1$ you get: $P\in I$ for all $P$, in other words $$R[X]\subset I$$

3. An ideal is a subset of the ring. Therefore, $$I\subset R[X]$$

Altogether, $I=R[X].$