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    Fundamental Math

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    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 Zp 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 Zp.

    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.

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    https://brainmass.com/math/algebra/fundamental-math-algebraic-number-theory-507582

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    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.

    $2.19

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