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Let $A=R[X_1,\dots,X_n]$ be the polynomial ring in n variables over an integral domain R. Denote by $A_d$ the set of all homogenous polynomials of degreed d (polynomials in A where only monomials $X_1^{v_1}\dots X_n^{v_n}$ of degree $d=v_1+\dots+v_n$). Prove that, as an R-Module, A is the direct sum of the submodules $A_d$, $d\in\mathbb{N}_0$.

We have $A_dA_e\subset A_{d+e}$. A polynomial $f\in A$ is homogenous of degree d iff the equation $f(tX_1,\dots , tX_n)=t^df(X_1,\dots X_n)$ holds in the polynomials ring $A[t]$. Clues?

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Is this really about splitting fields? – Gerry Myerson Oct 31 '12 at 5:41
Can you figure out the case $n=1$? Also, I agree with @GerryMyerson. This has nothing to do with splitting fields. The correct terminology is "graded ring". – Rankeya Oct 31 '12 at 6:58

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