$$I =\langle x^5+x^3+x^2-1, x^5+x^3+x^2 \rangle$$
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?
On
Ideals have several properties:
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 $$1\in I$$
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 $$R[X]\subset I$$
An ideal is a subset of the ring. Therefore, $$I\subset R[X]$$
Altogether, $I=R[X].$
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 $\gcd(x^5+x^3+x^2-1, x^5+x^3+x^2)$.