For a polynomial f over a ground field F, let [f] be the set of polynomials defi

Question

For a polynomial f over a ground field F, let [f] be the set of polynomials defined by [f] = {g: g = f mod p}.

a) Show that [f] = [g] (as sets) if and only if [f] and [g] have nonempty intersection

b) Define the operations + and * by:

[f] + [g] = [f+g], and [f]*[g]=[fg]

Prove that + and * are well-defined independent of f and g.

(Note: This is a legitimate question. The sets [f] and [g] can be represented by

different elements so that [f] = [f_1] and [g] = [g_1]. We must make sure that

[f] +[g] = [f_1]+[g_1] and [f]*[g]=[f_1]*[g_1].)

(c) Let FF be the set

FF = {[f]: f is a polynomial over the ground field F}

Show that FF is a field if p is a prime polynomial.

Explanation / Answer

Ordinals were introduced by Georg Cantor in 1883 to accommodate infinite sequences and to classify sets with certain kinds of order structures on them.[1] He derived them by accident while working on a problem concerning trigonometric series

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