The following shows how Lang proved a statement as a part of the primitive element theorem, which asserts that if a finite extension $E$ of a field $k$ is separable over $k$, then there exists an element $α∈E$ such that $E = k (α)$.
I am confused by the following argument, which is also highlighted in the picture above:
Then the elements $σ_{i}(α + cβ) (i = 1, ... , n)$ are distinct, whence $k(α + cβ)$ has degree at least $n$ over $k$.
I can't see how one can estimate the degree of $k(α+cβ)/k$ using the fact that the $n$ values $σ_{i}(α + cβ) (i = 1, ... , n)$ are distinct. I assume this is because the polynomial $$\prod_{i=1}^n (X - \sigma_{i}(\alpha+c\beta))$$ divides $Irr(\alpha+c\beta, k, X)$ (i.e. the irreducible polynomial of $\alpha+c\beta$ over $k$) but I can't either prove or disprove the latter statement. Can anyone help me to show how we can estimate the degree of the field extension $k(\alpha+c\beta)/k$ in this context, no matter if it is proved by using $Irr(\alpha+c\beta, k, X)$? Thanks in advance!