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I posted this question over at MHF to no avail, I'm not really sure what the ruling is on this kind of thing, I know this site was setup when MHF was down for a long time but you seem to still be active and a lot of clever people are still here so hopefully you don't mind taking a look at this for me

I have a couple of example questions that I'm trying to get my head around, a bit of guidance would be fabulous.

\(S:=\{f\in\mathcal{Q}[X,Y]\mid f(X,Y)=f(Y,X) \mbox{ and } \deg(f)\geq 0\}\)

1a: Give two polynomials that belong to \(S\).

1b: Find a finite basis of the ideal \((S)\) of \(\mathcal{Q}[X,Y]\) and justify your answer.

I then have the question where the questions are the same but based on this

\(S:=\{f\in\mathcal{Q}[X,Y]\mid f(X,Y)=-f(Y,X)\}.\)