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Unique Factorization Domains

Lets consider the number 2. Looking for a factorization of 2 over $ latex \mathbb{Z}$ we have 2 \cdot 1. Is this factorizartion unique ? If we are looking over another domain like $latex \mathbb{Z}[\sqrt{-6}]$ , then how easy is to find all the posible factorizations of 2 ? If we find one is it indeed unique?

2=(a+b\sqrt{-6})(c+d\sqrt{-6}) ,a,b,c,d \in \mathbb{Z}  considering the norm of 2 : N(2)=N((a+b\sqrt{-6}))(c+d\sqrt{-6}). But N(2)=(2+0\sqrt{-6})(2-0\sqrt{-6})=4.Taking norms for both sides we have :4=(a^2+6b^2)(c^2+6d^2) in $latex \mathbb{Z}$. On the RHS the factors can be a) either both of them 2 ,b) one of them 4 and the other 1.

a) It’s easy to so that the equation 2=x^2+6y^2 has no integer solutions (just consider (x,y)mod3).

b) 1=x^2+6y^2 \rightarrow 1=1+06 , which gives the factorization of norms: 4=4\cdot1 which is trivial factorization on $latex \mathbb{Z\sqrt{-6}}$ (we are looking for non-trivial ones).

One can read a little more diophantine_eq and how the integer equation 2x^3=y^2+1 can be solved over a unique factorization domain (UFD) such as  \mathbb{Z[i]}.

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Galois #1 (FixG)

Let E a field , the group of all automorfisms is AutE. If F is a subset of E , then one automorfism that leaves all the elements of F invariant ( \sigma (a)= a  , for every a \in F ) , is called F-automorfism. The set of all F-automorfism is called  G(E,F) and is a sub-group of the group of AutF.

Let G a sub-group of AutE of the field E. We call FixG the set of all the invariant elements of E according to G-elements , thus :

FixG = {a | a \in E , and \sigma (a)= a , for every \sigma \in G }.

The set FixG is a field , sub-field of E and is called fixed field of the group G.The fixed field of G(E,F) contains F and does not necessarily is the same field. 

more on FIxG later , we will see some interesting applications with respect to the roots of a polynomial  over a field.  

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Ada Lovelace day – women in science and technology

Today is Ada Lovelace day . Ada Lovelace Day is an international day celebrating the achievements of women in science, technology, engineering and maths by encouraging people around the world to talk about the women whose work they admire. This international day of celebration helps people learn about the achievements of women in STEM, inspiring others and creating new role models for young and old alike.

Who is Ada Lovelace ?

Ada Lovelace, was an English mathematician and writer .She worked with the inventor and engineer  Charles Babbage‘  who is considered “father of the computer” , on the  project of  the mechanical general-purpose computer, the analytical engine . Her notes on the engine include what is recognised as the first algorithm intended to be processed by a machine; thanks to this, she is sometimes considered the world’s first computer programmer , which is quite impressive if you think she lived in the 19th century .

Ada Lovelace died of cancer at 36, a few short years after the publication of “Sketch of the Analytical Engine, with Notes from the Translator”.

The Analytical Engine remained a vision, until Lovelace’s notes became one of the critical documents to inspire Alan Turning’s work on the first modern computers in the 1940s.

The programming language Ada was named after Ada Lovelace .

you can read more about her  here

 

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it looks like I’m gonna keep following posting about 2 things : CFD and abstract algebra when I’ve done “my homework”  .Especially for the 2nd one there maybe categories about Galois theory most of them in an arbitrary order , but i might release a chain of post with evolutionary results , so if someone reads this (tutorial)-chain , might begin with Galois#1 , and follows until the last Galois#n , will have a clue of Galois theory .Of course Galois#n will probably be the great theorem of Galois …

 

 

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