# Holy Splitting Fields, Math-man!

Algebra Level 5

Suppose $$p\in K[x]$$ is a polynomial of degree six, where $$K$$ is a field. What is the smallest upper bound we can obtain for the degree of the splitting field extension $$K(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\alpha_6):K$$, with $$E_{\alpha_i}(p)=0$$, with this information?

For the sake of clear notation,

$$\displaystyle E_\alpha: K[x]\to K[\alpha], \sum_{0\leq i\leq n}\lambda_ix^i\mapsto \sum_{0\leq i\leq n}\lambda_i \alpha^i$$,

called the evaluation map, is a ring homomorphism from $$K[x]$$ to $$K[\alpha]$$. As a fun little tidbit that might be helpful, because each $$\alpha_i$$ in this case is trivially algebraic over $$K$$, $$K[\alpha_i]=K(\alpha_i)$$.

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