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# Fundamental Math

See the attached file.

1) Compute the Cayley tables for the additive group Z and for the multiplicative group
Z of non-zero elements in Z.

2) Let G be a group written additively. Recall that the order of an element a is the
minimal natural number n such that na = 0. If such n does not exits then one says that
the order of a is infinity.

i) Find the order of the following elements 2; 3; 5; 6 2 Z12.

ii) If G is a group written multiplicatively, the order of an element a is the minimal
natural number n such that an = 1. Find the order of the elements

iii) Find the order of the following elements...

3) i) Let G be a group written additively. An element a of a group G is called a generator
if any element x 2 G has the form x = na for some integer n. For example ????1 and 1 are
generators of Z, while Q has no generators at all. Find all generators of the group Z12.
ii) In multiplicative notation, an element a of a group G is called a generator if any
element of G can be written as a power of a. Carl Friedrich Gauss proved that for any
prime p the group Zp has a generator. Verify this statement for all primes 17 giving
explicitly a generator of the group Zp in each case.

Remark. Can you see any regularity among these generators for di erent primes? Probably not. A conjecture of Artin (which is still open) claims that if a is an integer which is
not a perfect square there are in nitely many primes p for which a is a generator in Zp.

4) i) Let G be a group written multiplicatively. For any element a 2 G, consider the
map fa : G ! G given by fa(x) = ax. Prove that fa is always a bijection.

#### Solution Summary

The response solves various problems in algebraic number theory, involving computing multiplicative orders of elements of Z_p and verifying bijective maps and quadratic reciprocity.

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