Math 324 homework help Alberta university

Question - Math 324 Assignment 5 (due October 15)

1.(6) Consider the sum ∑ ω ω k, where ω varies over all n complex roots of x n − 1. Show
that this sum equals n if n di vides k, and equals 0 otherwise.

2.(6) Show that the greatest common divisor of 4k + 6 and 6k + 7 is 1, for all integers k.

3.(5) (9.1, #12) Let a, b, and m ≥ 1 be integers with a, b both relatively prime to m. Show
that ord
m(ab) = ordm(a)â‹…ordm(b) if ordm(a) and ordm(b) are relatively prime.

4.( 2+2+2 ) Determine all primitive roots mod p, when p = 11, 13, and 17.

5.( 5) Suppose that r and s are both primitive roots mod m. Show that
indr(a) ≡ indr(s) ⋅inds(a) mod φ(m) holds for all a wit ...Read More h (a, m) = 1.

6.( 6) (9.4, #4) Determine all c mod 13 for which cx4 ≡ 2 mod 13 has a solution x.

7.( 6) (9.4, #10) Show that there are infinitely many primes p ≡ 1 mod 8. Hint: Following
Euclid’s argument, start with a hypothetical list p
1, p2, …, ps of all of them, but now form
Q = (2p
1p2…ps)4 + 1. Recall (from 9.4, #9 or lecture 16 ) that every odd prime p for which
x4 ≡ − 1 mod p has a solution x is necessarily ≡ 1 mod 8. ...Read Less

Solution Preview - ¸) have period 2 Ï€, exactly, hence (using the notation of (D2.0)) e n(h) = 1 ⇔ h ≡ 0 mod n follows from (D2.0e)). By the division algorithm and (D2.0d), {e n(h): 0 ≤ h < n} = {en(1)h: 0 ≤ h < n} are all n of the different en(h)’s, all of which have (e n(1)h)n = en(1)hn = en(n)h = 1h = 1. Thus these are all of the ω’s and our s

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