Ian Chen 發問時間： 科學數學 · 6 年前

# 我有數學的問題~急x)

1. let f and g be polynomials over a field what is the degree of the polynomial f(g(x))?

2. does the set of all nonzero polynomials over a field form a group with respect to multiplication?

please steps by steps

### 1 個解答

• 6 年前
最佳解答

1. We denote by F the field considered. We suppose that the degree of f is p and that of g is q, with (p,q)∈N^2.　(N={0,1,2,....})

Let f(x)=a0+a1*x+...+ap*x^p with (a0,...,ap)∈F^p and ap≠0

g(x)=b0+b1*x+...+bq*x^q with (b0,...,bq)∈F^q and bq≠0

then f(g(x))=ap(b0+b1*x+...+bq*x^q)^p+.....+a1(b0+b1*x+...+bq*x^q)+a0

=ap*bq^p*x^pq+h(x) where h is a polynomial whose degree is inferior to pq.

Because ap*bq^p is nonzero (since ap≠0 and bq≠0), we can conclude that the degree of f(g(x)) is pq, which is also deg(f)*deg(g).

Therefore, deg(f o g)=deg(f)*deg(g) .

2.Let's take the polynomial f(x)=x for example. (It's a nonzero polynomial)

There is no polynomial g such that fg=1, where 1 is the identity element for the group.

So, the set of all nonzero polynomials over a field DOESN'T form a multipicative group.

參考資料： Myself I study maths in France