Let K be a field, let $T = K[X\_1, X\_2,...]$ be a polynomial ring, let $R=K[X\_1^{2}, X\_1X\_2,..,X\_i X\_j,..]$, and let $L = Frac(R)$ = field of fractions of R. How can we prove that $R =T \cap L$ ?



question about a particular Polynomial ring
Consider the involution $\sigma $ of $T$ which maps each $X\_i$ to $-X\_i$; the $\sigma $-invariant subring of $T$ is $R$. Similarly $\sigma $ extends to the field of fractions $E$ of $T$, and the $\sigma $-invariant subfield of $T$ is $L$. Thus $T\cap L=T\cap E^{\sigma }=T^{\sigma }=R$.


