Let F be a field of characteristic 0. Let f,g be irreducible polynomials over F. Let u be root of f, v be root of g; u,v are elements of field extension K/F. Let F(u)=F(v).(adsbygoogle = window.adsbygoogle || []).push({});

Prove (with using basic polynomial theory only, without using linear algebra and vector spaces):

1) deg f = deg g (deg f is degree of the polynomial f)

2) any element of F(u) is a root of some irreducible polynomial h over F

3) deg h divides deg f (h is as in 2))

Are the propositions 1),2),3) valid for any field F, not only for fields with char F = 0?

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# Field extension, properties,proof

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