\text { Find a monic, irreducible polynomial } m(x) \in \mathbb{Z}_{2}[x] \text { such that } \mathrm{Z}_{2}[x] /(m(x)) \simeq \mathbb{F}_{8} \text { By part (a) and Kronecker's theorem, we know

} \mathrm{Z}_{2}[x] /(m(x)) \simeq \mathrm{Z}_{2}[\beta] \simeq \mathrm{F}_{8} \text {, where } x+(m(x)) \longmapsto \beta \text { Show } \mathbb{Z}_{2}\left[\beta^{2}\right]=\mathbb{Z}_{2}[\beta] \text {. Hint: you can use Tower Law. }