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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