我有數學的問題~急x)
1. give an example of a field F and an irreducible polynomial f over F such that f(g(x)) is reducible over F where g ∈ F[x]。
2. let F be a field and let f,g ∈ F[x] be polynomials of positive degrees. if f =gd+r with deg(r)<deg(g)or a=0 find deg(d) in terms of the degrees of f and g。
please steps by steps
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1. Let F=Q, the set of rational numbers and f=x^2+x+1.
then f is irreducible over F cause its roots are irrational.
Now, let g=x^2.
Then f(g(x))=x^4+x^2+1=(x^2+1)^2-x^2=(x^2+x+1)(x^2-x+1), which proves that f o g is reducible over F.
2.
I think that the "a" in the second line should be "r".
In this case, we take the degree of each side, we obtain deg(f)=deg(gd+r).
then deg(gd+r)=deg(gd) because deg(r) is strictly inferior to deg(g)
and deg(gd)=deg(g)+deg(d) by a simple proof.
finally we get
deg(d)=deg(f)-deg(g).
是台灣的嗎?還是亂碼都看不懂><