Splitting Fields: Anderson and Feil, Theorem 45.5

[Math Amateur]

I am reading Anderson and Feil - A First Course in Abstract Algebra.

I am currently focused on Ch. 45: The Splitting Field ... ...

I need some help with some aspects of the proof of Theorem 45.5 ...

Theorem 45.5 and its proof read as follows:

View attachment 6680
View attachment 6681In the above text from Anderson and Feil we read the following:

"... ... Now \(\displaystyle \alpha\) and \(\displaystyle \beta \) are roots of irreducible polynomials \(\displaystyle f, g \in F[x]\) ... ... Now, we are just given that \(\displaystyle \alpha\) and \(\displaystyle \beta\) are algebraic elements of a field \(\displaystyle F\) ... ... how, exactly, do we know that they are roots of irreducible polynomials in \(\displaystyle F[x]\) ... .,.. ?
"( NOTE: A&F's definition of algebraic over \(\displaystyle F \) does not mention irreducible polynomials but says:

"If \(\displaystyle E\) is an extension field of a field \(\displaystyle F \) and \(\displaystyle \alpha \in E\) is a root of a polynomial in \(\displaystyle F[x]\), we say \(\displaystyle \alpha\) is algebraic over \(\displaystyle F\). ...)

Hope someone can help ...

PeterEdit: Hmm ... wonder if Kronecker's Theorem has something to do with irreducible polynomials entering this discussion ...

 

[jvicens]

Hello Peter,

Thank you for reaching out for help with Theorem 45.5 in Anderson and Feil's A First Course in Abstract Algebra. The proof of this theorem relies on the fact that \alpha and \beta are roots of irreducible polynomials in F[x], and I can understand your confusion as to how we know this.

First, let's clarify the definition of an algebraic element. As you mentioned, A&F define an element \alpha \in E as algebraic over F if it is a root of a polynomial in F[x]. This means that there exists some polynomial f \in F[x] such that f(\alpha) = 0. In other words, \alpha is a solution to the polynomial equation f(x) = 0. This is what it means for an element to be algebraic over a field.

Now, let's look at the definition of a splitting field. A splitting field of a polynomial f \in F[x] is an extension field E of F such that f factors completely into linear factors in E[x]. In other words, all the roots of f are contained in E. This is where Kronecker's Theorem comes into play. This theorem states that every polynomial f \in F[x] has a splitting field. So, in our case, since \alpha and \beta are algebraic elements of F, there exists a polynomial f \in F[x] such that f(\alpha) = f(\beta) = 0. This means that both \alpha and \beta are roots of f, and therefore, they are contained in the splitting field of f. Since the splitting field is an extension of F, we can conclude that \alpha and \beta are roots of irreducible polynomials in F[x].

I hope this helps clarify your question. If you need further assistance, please don't hesitate to ask. Keep up the good work with your studies!