Homework4.md

Homework 4

Some modular arithmetic

1. Working with the following set of Integers S = {0,1,2,3,4,5,6}

What is this

a) 4 + 4 = (4 + 4) % 7 = 1

b) 3 x 5 = (3 * 5) % 7 = 1

c) what is the inverse of 3 under

i) addition

To find the additive inverse of 3 in modulo 7 arithmetic, fist we need to find a number $x$ such that $(3+x)\mathrm{mod}7=0$.

The additive inverse of $3$ is therefore $\boxed{{\displaystyle 4}}$, since $3+4=7$ and 7 is equivalent to 0 modulo 7.

ii) multiplication

The multiplicative inverse of a number $a$ modulo $m$ is a number $b$ such that $a\cdot b\mathrm{mod}m=1$. For number 3 modulo 7, we need to find a $b$ such that $3\cdot b\mathrm{mod}7=1$.

The multiplicative inverse for 3 modulo 7 is $\boxed{{\displaystyle 5}}$, as $3\cdot 5=15$ and $15\mathrm{mod}7=1$.

2. For S = {0,1,2,3,4,5,6} Can we consider 'S' and the operation '+' to be a group?

To determine if a set together with a specific operation forms a group, we must ensure that the set satisfies four group axioms: Closure, Associativity, Identity Element, Invertibility

Given the set $ S = {0, 1, 2, 3, 4, 5, 6} $ and assuming the operation is addition modulo 7, we can check each of the group axioms:

Closure: Addition of any two elements in $ S $ will result in another element in $ S $ when taken modulo 7, thus $ S $ is closed under addition modulo 7.

Associativity: Addition is associative under regular integer arithmetic, and this property carries over to addition modulo 7.

Identity Element: The identity element for addition is $ 0 $ since adding $ 0 $ to any element $ a $ does not change $ a $, and $ 0 $ is in the set $S$.

Invertibility: Every element $ a $ has an additive inverse $ b $ such that $ a + b \mod 7 = 0 $. Since our set is a complete residue system modulo 7, each element indeed has an inverse within the set.

Considering these properties, $ S $ forms an additive group modulo 7.

3. What is -13 mod 5? 2

4. Polynomials

For the polynomial $(x^3-x^2+4x-12)$

Find a positive root

By using the Ruffini method and this Ruffini code writen in rust, we get that a rational root is: 2

What is the degree of this polynomial? 3