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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:

In the above text from Anderson and Feil we read the following:

"... ... Now ##\alpha## and ##\beta## are roots of irreducible polynomials ##f, g \in F[x]## ... ...

Now, we are just given that \alpha and \beta are algebraic elements of a field ##F## ... ... how, exactly, do we know that they are roots of

"( NOTE: A&F's definition of algebraic over ##F## does not mention irreducible polynomials but says:

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

Hope someone can help ...

Peter

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

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:

In the above text from Anderson and Feil we read the following:

"... ... Now ##\alpha## and ##\beta## are roots of irreducible polynomials ##f, g \in F[x]## ... ...

Now, we are just given that \alpha and \beta are algebraic elements of a field ##F## ... ... how, exactly, do we know that they are roots of

*polynomials in ##F[x] ##... .,.. ?***irreducible**"( NOTE: A&F's definition of algebraic over ##F## does not mention irreducible polynomials but says:

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

Hope someone can help ...

Peter

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