Ian Chen 發問時間: 科學數學 · 6 年前


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

    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