I have $f = XY+Y$ and $g = X^2 +1$ in $\mathbb{Q}[X,Y]$. Let $I =\langle f,g\rangle$ be the ideal in $\mathbb{Q}[X,Y]$ generated by $f$ and $g$.
I have that $in_{<}(f) = XY$ and $in_{<}(g) = X^2$. I also have that $S(f,g) = XY-Y$ since $\operatorname{lcm}(X^2, XY) = X^2Y$.
Now I need to show that $Y \in I$.
I can't really see how I'm supposed to do that. I guess that I need to show that $Y$ is a linear combination of $f$ and $g$, but I'm not entirely sure how.
The next part is showing that $(Y,g)$ is a Gröbner basis. That I can do - just after I have shown that $Y \in I$.
And the next part is:
Let $h \in \mathbb{Q}[X,Y]$. Show that there exist unique numbers $a,b \in \mathbb{Q}$ with $h-(a+bX) \in I$.
How would I go about doing this?
This is not homework. It is example-assignments for the exam which we can use to practice. So getting an idea of how to solve this kind of problems would help me a lot. Thanks!
I think, in general, you should find a Gröbner basis first! However, in this particular case it's not strictly necessary as @LutzL remarked: $Xf-Yg=XY-Y\in I$, so $f−(XY−Y)=2Y\in I$, and thus you obtain $(1/2)(2Y)=Y\in I$.
Then use the Buchberger's algorithm and get exactly what you said: $XY+Y,X^2+1,Y$ is a Gröbner basis. (I have assumed that the monomial order is lex and $X>Y$.) Now take an arbitrary $h\in\Bbb Q[X,Y]$ and divide it to the Gröbner basis. We get $$h(X,Y)=(XY+Y)h_1(X,Y)+(X^2+1)h_2(X,Y)+Yh_3(X,Y)+r(X,Y)$$ where every monomial of $r(X,Y)$ is not divisible by any of $XY, X^2, Y$. Then how look the monomials of $r(X,Y)$? In the end use that $r$ is unique since you divided $h$ by a Gröbner basis.