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 $R =T \cap L$ ?
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1 Answer
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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 $T\cap L=T\cap E^{\sigma }=T^{\sigma }=R$.
