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vVm2&Z%0tG$|EU|?*8M-8r=1Ilub-`tUyu*nfkYe(mXehs;o;FR&?NaUH%Ygy diff --git a/seminarexercises.tex b/seminarexercises.tex index d1c13d5..0e4c29a 100644 --- a/seminarexercises.tex +++ b/seminarexercises.tex @@ -42,12 +42,8 @@ \end{solution} \begin{solution}{1.28}\ - - The quotient set of the equivalence relation in Example 1.27 is - $$ - X / \sim = \{[(x_0,y_0)], [(x_1, y_1)], \ldots, [(x_n, y_n)]\} - $$ - Yes, it is isomorphic to the cosets of the \emph{nth} roots of unity, which are $\mathbb{G}_n = \{w_k\}^{n-1}_{k=0}$, where $w_k=e^{\frac{2 \pi i}{n}}$. + \emph{(WIP)}\\ + It is isomorphic to the cosets of the \emph{nth} roots of unity, which are $\mathbb{G}_n = \{w_k\}^{n-1}_{k=0}$, where $w_k=e^{\frac{2 \pi i}{n}}$. \end{solution} \begin{solution}{2.2}\

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That is, there are one or more primes $p_1, \ldots, p_r$ such that $n=p_1 p_2 \cdots p_r$. +\end{theorem} + +\begin{theorem}[Unique factorization] + Suppose $n$ can be factored into positive primes in two ways, namely, + $$n= p_1 \cdots p_r = q_1 \cdots q_t$$ + Then $r=t$, and the $p_i$ are the same numbers as the $q_j$ except, possibly, for the order in which they appear. +\end{theorem} + +From the last two theorems: every integer $m$ can be factored into primes, and the prime factors of $m$ are unique (except for the order). + + +\section{Elements of number theory} + +\begin{theorem}[Little theorem of Fermat] + Let $p$ be a prime. Then, + $$a^{p-1} \equiv 1 \pmod p, \forall a \not\equiv 0 \pmod p$$ + \\ + So, by taking $a^{p-2} \cdot a \equiv 1 \pmod p$, where $a^{p-2} \equiv a^{-1} \pmod p$ (the inverse modulo p), we see that $a^p \equiv a \pmod p, \forall a \in \mathbb{Z}$, so $a^p - a$ is a multiple of $p$. +\end{theorem} + +\begin{theorem}[Euler's $\phi$ function] + \emph{Euler's $\phi$ function} describes the number of integers in $\mathbb{Z} / n \mathbb{Z}$ which are relatively prime (coprime) to $n$. +\end{theorem} + +\begin{theorem}[Euler's theorem] + If $a$ and $n$ are relatively prime, $a^{\phi(n)} \equiv 1 \pmod n$. +\end{theorem} + + +\section{Polynomials} + +\begin{definition} + Let $A$ be a commutative ring with unity, and $x$ an arbitrary symbol. Every expression of the form + $$a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$$ + is called a \emph{polynomial in $x$ with coefficients in $A$}, or more simply, a \emph{polynomial in $x$ over $A$}. +\end{definition} + The expressions $a_k x^k$, for $k \in \{ 1, \ldots, n \}$, are called the \emph{terms} of the polynomial, being $a_n x^n$ the \emph{leading term}, and $a_0$ the \emph{constant term}. + The $a_k$ are called the \emph{coefficients} of $x^k$, being $a_n$ the \emph{leading coefficient}. And the \emph{degree} of a polynomial $a(x)$ is the greatest $n$ such that the coefficient of $x^n$ is not zero. + The polynomial whose leading coefficient is equal to $1$ is called \emph{monic}. + +\begin{theorem}[Division algorithm for polynomials] + If $a(x)$ and $b(x)$ are polynomials over a field $F$, and $b(x) \neq 0$, there exist polynomials $q(x)$ and $r(x)$ over $F$ such that + $a(x) = b(x) q(x) + r(x)$ and + [$r(x) = 0$ or $\deg r(x) < \deg b(x)$]. +\end{theorem} + +\begin{theorem} + Any two nonzero polynomials $a(x), b(x) \in F[x]$ have a $\gcd d(x)$. Furthermore, $d(x)$ can be expressed as a \emph{linear combination} + $$d(x)= r(x) a(x) + s(x) b(x)$$ + where $r(x), s(x) \in F[x]$. +\end{theorem} + +\begin{theorem}[Factorization into irreducible polynomials] + Every polynomial $a(x)$ of positive degree in $F[x]$ can be written as a product + $$a(x) = k p_1(x) p_2(x) \cdots p_r(x)$$ + where $k$ is a constant in $F$ and $p_1(x), \ldots, p_r(x)$ are monic irreducible polynomials of $F[x]$. +\end{theorem} + +\begin{theorem}[Unique factorization] + If $a(x)$ can be written in two ways as a product of monic irreducibles, say + $$a(x) = k p_1(x) \cdots p_r(x) = l q_1(x) \cdots q_s(x)$$ + then $k=l$, $r=s$, and $p_i(x) = q_j(x)$. +\end{theorem} + +\begin{theorem} + $c$ is a root of $a(x)$ iff $x - c$ is a factor of $a(x)$. +\end{theorem} +\begin{theorem} + If $a(x)$ has distinct roots $c_1, \ldots, c_m$ in $F$, then $(x-c_1)(x-c_2)\cdots(x-c_m)$ is a factor of $a(x)$. +\end{theorem} + +\begin{theorem} + If $a(x)$ has degree $n$, it has at most $n$ roots. +\end{theorem} + + +\framebox{WIP: covered until chapter 26, work in progress.} \end{document} diff --git a/number-theory.sage b/number-theory.sage index 68e42d6..1340123 100644 --- a/number-theory.sage +++ b/number-theory.sage @@ -32,6 +32,12 @@ def gcd(a, b): return g*b +def gcd_recursive(a, b): + if mod(a, b)==0: + return b + return gcd_recursive(b, mod(a,b)) + + # Extended Euclidean algorithm # Inputs: a, b # Outputs: r, x, y, such that r = gcd(a, b) = x*a + y*b @@ -54,6 +60,13 @@ def egcd(a, b): return d, x, y +def egcd_recursive(a, b): + if b==0: + return a, 1, 0 + + g, x, y = egcd_recursive(b, a%b) + return g, y, x - (a//b) * y + # Inverse modulo N, using the Extended Euclidean algorithm def inv_mod(a, N): g, x, y = egcd(a, N) diff --git a/number-theory_test.sage b/number-theory_test.sage index f6de808..6709f3a 100644 --- a/number-theory_test.sage +++ b/number-theory_test.sage @@ -18,9 +18,12 @@ assert crt(a_i, m_i, M) == 42 assert gcd(21, 12) == 3 assert gcd(1_426_668_559_730, 810_653_094_756) == 1_417_082 +assert gcd_recursive(21, 12) == 3 + ##### # Extended Euclidean algorithm tests assert egcd(7, 19) == (1, -8, 3) +assert egcd_recursive(7, 19) == (1, -8, 3) ##### # Inverse modulo N tests diff --git a/seminarexercises.pdf b/seminarexercises.pdf index d693648b4af5717500707dbd528c3df3521c29b6..cd6715b2a2884d419bec84161b13a6da7d2eab90 100644 GIT binary patch delta 53894 zcmV(~K+nIZ!3UhX2aqEKI50IJld+#Cf9+XGZ`(!`zWZ13NCDww_N9dn36KN{dT0>= zIk-M3Hf;&9MMp~=U;5kohQnos9*UGB%Q^H!iG1_;mhY{^$#sI04=1?i`gVHq`aKm1 zQHT=3lj&7Lxj@YDL=tLe2wX>8o-_Y)1LzOJ^j)dR4gDy5iGd%6+i^?S|}gNy`jz$GJ0AOcckOd>67 zDTGAGhh{0f69qh4UfN@Yff@V*B}*P8Jd>P(q&WQ}-#BFm@s2W#ga)$m;FTlr zSqsp*e*8BUIK~2L#cP^bj0Kg-f8#+W8C5zxjj#eA5z)|YW4sh@-t`KM#8|qS=gSB$ zs*%#9roE2wBH+C}bbGpa#DICEZ=T&Iq6A?^51^J9Ff;RIs!Y8uViM63#-T*Shq^+H zdhCddZ~@O7xx>;1QPp^Jv3Byz?1iUS+vSA~nhx$WiWvf1xZsfPD2)ckf0qeFTke=% z%$Ak5dG7dc=Ce)pthl`c-{`bhFRCzwyTw&4$AqL#l(oCl2)YK-*>e|**=F(21b$V# 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