prof: usualy at IM 223
Excersise Classes: Julian Danner (IM 207)
An abelian group is a set $( G )$ equipped with a binary operation $( + )$ (often called addition) that satisfies the following properties:
A commutative group (also known as an abelian group) is a group $( G, + )$ where the group operation $( + )$ satisfies the commutative property:
This means that the order in which elements are combined does not affect the result.
A ring is a set $( R )$ equipped with two binary operations, typically called addition $( + )$ and multiplication $( \cdot )$, such that:
Additive abelian group: $( R, + )$ is an abelian group. This means:
Multiplicative closure and associativity:
Distributive properties:
A ring may or may not have a multiplicative identity (an element $( 1 \in R )$ such that for all $( a \in R )$, $( a \cdot 1 = a )$). If it does, the ring is called a unital ring or ring with unity.
A commutative field (or simply a field) is a set $( F )$ equipped with two binary operations, addition $( + )$ and multiplication $( \cdot )$, such that:
Additive abelian group: $( F, + )$ is an abelian group. This means:
Multiplicative abelian group: $( F \setminus {0}, \cdot )$ is an abelian group. This means:
Distributive property:
A field is essentially a commutative ring with unity where every nonzero element has a multiplicative inverse.
A polynomial ring is a ring $( R[x] )$ formed by the set of all polynomials with coefficients in a ring $( R )$ and indeterminate $( x )$. The elements of $( R[x] )$ are expressions of the form:
$ f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 $
where $( a_0, a_1, \dots, a_n \in R )$, $( n \geq 0 )$, and $( x )$ is an indeterminate.
The operations of addition and multiplication in $( R[x] )$ are defined as follows:
$ f(x) = a_n x^n + \dots + a_0, \quad g(x) = b_m x^m + \dots + b_0 $
their sum is:
$ f(x) + g(x) = (a_n + b_n)x^n + \dots + (a_0 + b_0) $
$ f(x) = a_n x^n + \dots + a_0, \quad g(x) = b_m x^m + \dots + b_0 $
their product is:
$ f(x) \cdot g(x) = \sum_{i=0}^{n+m} \left( \sum_{j+k=i} a_j b_k \right) x^i $
where the coefficients are obtained by summing the products of terms with matching degrees.
The polynomial ring $( R[x] )$ inherits the properties of the coefficient ring $( R )$. If $( R )$ is commutative, then $( R[x] )$ is also commutative. If $( R )$ has a unity, then $( R[x] )$ also has a unity, which is the constant polynomial $( 1 )$.
The ring $Z/nZ$ is a field if and only if $( n )$ is prime. This is because the elements of $Z/nZ$ are the equivalence classes of integers modulo $( n )$, and the multiplicative inverses exist only when $( n )$ is prime. In this case, every nonzero element has a multiplicative inverse in the field.
__In this lectures main fields are $Q$ and $Fp$ fields__
$x_1$, $x_2$, $x_3$, ..., $x_n$ - some indeterminate variables
$x_1^{\alpha_1}$, $x_2^{\alpha_2}$, ..., $x_n^{\alpha_n}$ - terms or monomials
A commutative ring is a ring $( R )$ in which the multiplication operation $( \cdot )$ is commutative. This means that for all $( a, b \in R )$, the following property holds:
In other words, the order in which two elements are multiplied does not affect the result. A commutative ring still satisfies all the other properties of a ring, including the distributive properties and the existence of an additive identity and additive inverses.
A commutative ring is called a polynomial ring when it consists of polynomials with coefficients in a commutative ring $( R )$ and one or more indeterminates $( x_1, x_2, \dots, x_n )$.
For example, the polynomial ring $( R[x] )$ is formed by the set of all polynomials with coefficients in $( R )$ and a single indeterminate $( x )$. Similarly, $( R[x_1, x_2, \dots, x_n] )$ represents the polynomial ring with multiple indeterminates.
The key properties of a polynomial ring are:
A commutative ring becomes a polynomial ring when its elements can be expressed as polynomials over a base ring $( R )$ with specified indeterminates.
let K be a field and $A\in Mat_n(K)$. The minimal polynomial of $A$ is the monic polynomial of least degree such that $p(A)=0$. The minimal polynomial is unique and divides any polynomial that annihilates $A$. The roots of the minimal polynomial are the eigenvalues of $A$.
Solution:
Calculate $K_A(z) = det(zI-A)$, where $I$ is the identity matrix. The roots of $K_A(z)$ are the eigenvalues of $A$. The minimal polynomial is the monic polynomial of least degree that divides $K_A(z)$ and has the same roots as $K_A(z)$.
And its prome factorization is $K_A(z) = $p_1(z)^x_1$ ... $p_k(z)^x_k$
Remove vectors until we dont get a zero vector.
a = $\sqrt{2}$ + $^3\sqrt{3}$ + $i \in C$. Find the minimal polynomial of $a$ over $Q$, such that $p(a)=0$.
Solution:
Haven't write down
3-coloring of a graph
A graph $G = (V, E)$ consists of a set of vertices V and a set of edges E. A 3-coloring of G is an assignment of colors to the vertices such that no two adjacent vertices have the same color. The chromatic polynomial $P(G, k)$ counts the number of ways to color the graph with k colors.
$E <= V*V$
lets call colors $\overline{0},$ $ \overline{1}$, $\overline{2}$
Model the task using polinomials.
$V = {v_1, v_2, \dots, v_n}$ $< - >$ elements $x_1, x_2, \dots, x_n$ colors $< - >$ elements $\overline{0},$ $ \overline{1}$, $\overline{2}$ $\in$ $F_3$
Look at P = $F_3[x_1, x_2, \dots, x_n]$.
(1) each $v_i$ gets one color: $x_i(x_i - \overline{0})(x_i - \overline{1})(x_i - \overline{2}) = 0$ -> $x_i^3 - x_i$
a 3-coloring is a 0 of $< x_1^3 - x_1, \dots, x_n^3 - x_n >$
(2) if ($v_i, v_j$) is an edge, then $x_i \neq x_j$ -> $< x_i - x_j >$
$x_i^2 + x_i * x_j + x_j^2 = 0$ for all $(v_i, v_j) \in E$.
Goal: find a common zero $(a_1, a_2, \dots, a_n)$ of all polynomials.