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We study the ring of invariants for a finite dimensional representation $V$of the group $C_2$ of order 2 in characteristic $2$. Let $\sigma$ denote agenerator of $C_2$ and $\{x_1,y_1 \dots, x_m,y_m\}$ a basis of $V^*$. Then$\sigma(x_i) = x_i$, and $\sigma(y_i) = y_i + x_i$. To our knowledge, this ring(for any prime $p$) was first studied by David Richman in 1990. He gave a firstmain theorem for $(V_2, C_2)$, that is, he proved that the ring of invariantswhen $p=2$ is generated by $\{x_i, N_i = y_i^2 + x_iy_i, tr(A) | 2 \le |A| \lem\}$ where $A \subset \{0,1\}^m$, $y^A = y_1^{a_1} y_2^{a_2} \cdots y_m^{a_m}$and $tr(A) = y^A + (y_1+x_1)^{a_1}(y_2+x_2)^{a_2} \cdots (y_m+x_m)^{a_m}.$ Inthis paper, we prove the second main theorem for $(V_2, C_2)$, that is, we showthat all relations between these generators are generated by relations of typeI: $\sum_{I \subset A } x^I tr(A-I)$ and of type II: $tr(A) tr(B) = \sum_{L <I} x^{I-L} N^L tr(I-L+J+K) + N^I \sum_{L < J} x^{J-L} tr(L+K)$ for all $m$. Wealso derive relations of type III which are simpler and can be used in place ofthe relations of type II.

The Second Main Theorem Vector for the modular regular representation of $C_2$