Advanced Algebra Study Notes (1) Polynomials - Number Field

polynomial

1. Number range

Definition 1 Let $P$ is a collection of complex numbers, including 0 and 1. IfPPThe sum, difference, multiplication, product, and quotient of any two numbers in$P$ (these two numbers can also be the same) (the divisor is not 0) is still PPnumber in$P$$P$ is called anumber field.

For example:
1. The set of all rational numbers (denoted as Q)
2. The set of all real numbers (denoted as R)
3. The set of all complex numbers (denoted as C)

If the set of numbers PPThe result of doing a certain operation on any two numbers in$P$$In\; P$ , we say that the set of numbers$P$ is closedto this operation. Therefore, the definition of the number field can also be said that if a number set$P$ is closedfor addition, subtraction, multiplication and division (the divisor is not 0), then$P$ is called anumber field.

Example 1: All have the form
$$a+b\sqrt{2}$$
The number of (where $a,b$ is any rational number) to form a number field. Usually Q($\sqrt{2}$) to represent this number field.
Proof:
Since $a,b$ is a rational number, then$b\sqrt{2}$is also a rational number.
Rational numbers + rational numbers = rational numbers
Rational numbers - rational numbers = rational numbers
Clearly, the set of numbers Q( $\sqrt{2}$) contains 0 and 1, and it is closed for addition and subtraction.
For multiplicative closure:
$$(a+b\sqrt{2})(c+d\sqrt{2})=(ac+2bd)+(ad+bc)\sqrt{2}$$
因为$a,b,c,d$ are rational numbers, so$ac+2bd,ad+bc$ is also a rational number, which means$(a+b\sqrt{2})(c+d\sqrt{2})$ is still in Q($\sqrt{2}$), so Q( $\sqrt{2}$) is closed for multiplication.
For division closure:
Let $a+b\sqrt{2}\ne 0$ , so$a-b\sqrt{2}\ne 0$ (one one)indicates,
$$\begin{array}{rl}a+b\sqrt{2}& \ne 0\\ a-(-b\sqrt{2})& \ne 0\\ a-0& \ne b\sqrt{2}\\ a& \ne b\sqrt{2}\\ a-b\sqrt{2}& \ne 0\end{array}$$
而
$$\begin{array}{c}\begin{array}{rl}\frac{c+d\sqrt{2}}{a+b\sqrt{2}}& =\frac{(c+d\sqrt{2})(a-b\sqrt{2})}{(a+b\sqrt{2})(a-b\sqrt{2})}\\ & =\frac{ac-2b}{a^2-2b^2}+\frac{ad-2b}{a^2-2b^2}\sqrt{2},\end{array}\end{array}$$
因为$a,b,c,d$ is a rational number, so$a^2-2b^2$ is a non-zero rational number,$\frac{ac-2b}{a^2-2b^2}$,$\frac{ad-2bc}{a^2-2b^2}$is also a rational number. Then Q( $\sqrt{2}$) is closed for division.
In summary, Q( $\sqrt{2}$) is a number field.

Example 2 : Ownable or less table formula
$$\frac{a_0+a_{1}Pi+...+a_n{Pi}^n}{b_0+b_{1}Pi+...+b_m{Pi}^m}$$
The arrays form a number field, where n, m are any non-negative integers, $a_i,b_j(i=0,...,n;j=0,...,n;)$ condition.output
:
1.output:
$$\begin{gathered} \frac{a_0+a_{1}Pi+...+a_n{Pi}^n}{b_0+b_{1}Pi+...+b_m{Pi}^m}+\frac{c_0+c_{1}Pi+...+c_n{Pi}^n}{d_0+d_{1}Pi+...+d_m{Pi}^m}= \\ \frac{a_0+c_0+(a_1+c_1)p+...+(a_n+c_n)p^n}{b_0+d_0+(b_1+d_1)p+...+(b_m+d_m)p^m} \end{gathered}$$

Let $l,k$ is any non-negative integer,$a_l,b_k$is an integer
since $a_n，c_n$is an integer, then $a_n+c_n，b_m+d_m$is also an integer, then
in $l$ can always find an$a_l$, make $a_n+c_n=a_l$
in kkA bk b_kcan always be found in$k$$b_k$, so that $b_m+d_m=b_k$
And l, k, n, m are all non-negative integers and can be transformed into each other, so the closure of addition is established.

2. Closeness of subtraction: Addition in the same way.

3. 乘法动生性：
$$\begin{gathered} (\frac{a_0+a_{1}Pi+...+a_n{Pi}^n}{b_0+b_{1}Pi+...+b_m{Pi}^m})(\frac{c_0+c_{1}Pi+...+c_n{Pi}^n}{d_0+d_{1}Pi+...+d_m{Pi}^m})= \\ \frac{a_0{c}_0+(a_0{c}_1)p+...+\left(a_n{c}_n\right)p^n}{b_0{d}_0+(b_0{d}_1)p+...+\left(b_m{d}_m\right)p^m} \end{gathered}$$

Let $l,k$ is any non-negative integer,$a_l,b_k$is an integer
since $a_n，c_n$is an integer, then $a_n{c}_n，b_m{d}_m$is also an integer, then
in $l$ can always find an$a_l$, use $a_n{c}_n=a_l$
in kkA bk b_kcan always be found in$k$$b_k$, making $b_m{d}_m=b_k$
And l, k, n, m are all non-negative integers, which can be transformed into each other, so the closure of multiplication is established.

4. Closeness of division: observe that the formula is a fraction, and dividing by a fraction is equal to multiplying the reciprocal of this fraction, which is the same as multiplication, so the closure of division is also established.
In summary, this form is a number field.

Example 3 : The number set composed of all odd numbers is closed for multiplication, but not closed for addition and subtraction. $\sqrt{2}$All integer multiples of , form a number set, which is closed for addition and subtraction, but not for multiplication and division. Of course, the above two number sets are not number fields. Finally, we point out an important property of number fields.
All The number field of contains the rational number field as part of it.