Prove that for any positive integer n, a field F can have at most afinitenumber of elements of multiplicative order at most n.

Is there a theorem that says if a field F has order n, then F* has order n-1

9. Recall from Section 2 that a field F is called an ordered field if there exists a subset P of F (called the set of positive elements) such that (a) sums and products of elements in P are in P, and (b) for each element a in F, one and only one of the following possibilities holds: a e P, a = Prove that the field of complex numbers is not an ordered field. 0, - a e P.

Given finite field GF(16), can you perform arithmetic operations on the elements of the field as integers from 0 to 15 mod 16, such as: 5*6 mod 16 =14? Explain your answer.

Prove that any field is either of characteristic 0 or of characteristic p, where p is a prime.

If K is a subfield of the Galois field GF(p\power{n}), then there exists an integer m such that K contains p\power{m} elements and m is a divisor of n.

Let p be a prime number. For example, p could be 2,3,5,7, etc. We define a field Fp consisting of the elements {0, 1, 2, . . . , p − 1} with operations +, −, ×, ÷ considered only up to remainder when we divide by p. For example,in F5,2+4 = 6≡ 1,2−4 = −2 ≡ 3,and2×4 = 8 ≡3. SinceF5, 4 × 3 = 12 ≡ 2 (you can check that the multiplication of 4 with any other element in F5 is never 2), we say that 2 ÷ 4 ≡ 3 in F5.