While self studying algebra from Thomas Hungerford I have following question in Theorem 3.8 on page 286.
Hiw to rigorously prove that if f splits over K then $\sigma f $ over L?
Similarly, How to prove rigorously that if f has an irreducible factor g in F then $\sigma g $ would be irreducible in L[x]?
For both, I thought of assuming that $\sigma f $ doesn't splits over L and $\sigma g $ is reducible in G and trying to find a contradiction using that $\sigma$ : K $\to$ L is an isomorphism.
But I am not sure how to obtain that it will imply (intuitively I think following must be true) f doesn't splits over K and g is reducible ie how to use map $\sigma$ for it.
Can you please give a rigorous proof for these two?
I shall be really thankful.