# 2 questions in proof of Theorem 3.8 of Galois Theory in Hungerford Algebra

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.

How to rigorously prove that if $$f$$ splits over $$K$$ then $$σf$$ splits over $$L$$?
If $$f$$ splits over $$K$$, then $$f=a\prod_i(x-a_i),$$ with $$a,a_i\in K$$. Applying $$\sigma$$ we have $$\sigma f=\sigma(a)\prod_i(x-\sigma(a_i)),$$ with $$\sigma(a),\sigma(a_i)\in L$$ and so $$\sigma(f)$$ splits over $$L$$.
Similarly, How to prove rigorously that if $$f$$ has an irreducible factor $$g$$ in $$K$$ then $$σg$$ would be irreducible in $$L[x]$$?
Suppose on the contrary that $$\sigma g=h_1\cdot h_2$$, with $$h_1,h_2\in L[x]$$ then, because $$L\cong K$$, $$g=\sigma^{-1}(\sigma g)=\sigma^{-1}(h_1)\sigma^{-1}(h_2)$$ would be not irreducible in $$K[x]$$, in particular would be not irreducible in $$L[x]$$, because $$K\subseteq L$$.