The existence of the normal subgroups of GF($2^m$) under addition

Question

If we construct our GF($2^4$) with primitive polynomial $\alpha^4 + \alpha + 1 = 0$ where $\alpha$ is the primitive element in extended Galois field GF($2^4$). The elements of the GF($2^4$) generated by $\alpha$ as such $\{0, 1, \alpha, \alpha^2, \alpha^3,\alpha^4,\alpha^5,\alpha^6,\alpha^7,\alpha^8,\alpha^9,\alpha^{10},\alpha^{11},\alpha^{12},\alpha^{13},\alpha^{14}\}$. I am interested to know if there exist a subgroup under addition operation (not multiplication). More so if there is a "normal" subgroup under addition. One possible additive subgroup that I found is $\{0, 1, \alpha, \alpha^4\}$. Why there is little on additive property of GF($2^4$). All the literature talks about multiplicative (normal) subgroups etc. ?