2. no. For example, if m is the union of rational number set and single element set {√2}, then it is easy to find that m does not conform to the definition of domain. Use non-zero elements and √2 to add, subtract, multiply and divide.
3. That's right.
(1) Since the number field P contains nonzero elements A, a+a=2a∈P, a-a=0∈P, 2+2a=3a∈P, ..., aZ={na|n is an integer} is a subset of P, and aZ is infinite.
Furthermore, for non-zero element A, a/a= 1∈P, so all integers are ∈P. But a/b∈P, so the set of rational numbers is a subset of p. In other words, the set of rational numbers is the smallest number field.
(2) If the number field P is a finite set, let all its elements be {a 1, a2, ..., one}. Then, these elements are divided into non-negative numbers and negative numbers. Let the positive and negative parts be {a 1, ..., am} and the negative part be {am+ 1, ..., one}. Then, a 1+a2+...+am is a positive number greater than all a 1, ..., am and a 1+a2+...+am∈P have only n elements.
4. You probably wrote it wrong, "There are infinite fields".
Because it is proved in 3 that the rational number set Q is a subset of any number field, the number field is an infinite set.
(1) We consider the set A2={a+b√2|a, b∈Q}. It is easy to verify that the addition, subtraction, multiplication and division of any two numbers in this set are still elements in A2, and the division can be verified by rationalizing the denominator. Therefore, A2 is a domain.
Similarly, A3={a+b√3|a, b∈Q}, A5={a+b√5|a, b∈Q}, A6 = {a+b √ 6 | a, b ∈ q} ... Because the number that can't be opened is infinite, so is the number field.
Of course, the number field is not just above. The set of real numbers r is a number field. {a+b√2+c√3+d√6|a, B, C, d∈Q} is also a number field. Wait, wait, wait.
(2) Take an arbitrary number field F (such as a rational number set) and add a new element A to it, then F can be expanded into a new number field, which is denoted as F[a].
Similarly, F[a] can be expanded into F[a, b] by adding a new element b.
If you continue to expand, you can get an infinite number of fields.
3 and 4 are not contradictory. 3 means that the number field itself is an infinite set, but we know that the infinite set is not just a few, but an infinite number. And 4 just wants to find a set that satisfies the condition of the domain from this infinite set, and we will try to find it.
Number field, or field, is the learning content of abstract algebra (or modern algebra) in universities. Its basic contents are as follows:
The research object is an arbitrary set, not necessarily a set of numbers. But for convenience, let's take several episodes as examples.
Concept 1: group
The first letter of a nonempty set G(G (group), if its elements satisfy:
(1) Any element A, b∈G has AB ∈ G;
(2) Any element A, B, c∈G has A (BC) = (AB) C;
(3) Element E exists, so that for any element A in G, there is AE = EA = A；;
(4) For any element a∈G, there is an element A- 1, which is called the inverse of A, so AA-1= A-1A = E.
A set that satisfies this condition is called a group. The operations of (1)-(4) can be multiplication or addition. Among them, multiplication is called multiplication group and addition is called addition group.
Example: {0} is the simplest addition group. Because 0+0=0, 0+(0+0)=(0+0)+0, 0+0=0+0=0, the reciprocal of 0 is still 0.
{1} is the simplest multiplication group, and it is easy to verify these conclusions.
Integer set z is an additive group.
Obviously, as long as a group has addition, there must be subtraction (that is, adding inverse elements). As long as there is multiplication, there is division. However, there is only one operation. In a group, there is no need for the existence of addition (or multiplication) commutative laws. For example, we can forcibly think that 2+3 and 3+2 are not equal, which is a bit like a point in the coordinate system. If the commutative law holds, it is called a commutative group.
There is also "module", which is a special commutative group. It is so different from a group that it must be studied separately, just like the difference between a parallelogram and a quadrilateral.
Concept 2: Ring
If a nonempty set R (the first letter of a ring) is both an addition group (with additive commutative law) and a multiplication, and the multiplication satisfies the associative law and the distributive law (but the multiplication does not necessarily have constants and inverses), then R is called a ring.
There are addition, subtraction, multiplication and division in the ring, but division has not been defined.
For example, the simplest ring is {0}. The integer set z is a ring. All even numbers are rings.
Because a ring defines two sets of operations, it is neither an addition ring nor a multiplication ring.
Addition in a ring requires commutative law. If the multiplicative commutative law holds, it is called commutative ring. Otherwise it is called a non-commutative ring.
Concept 3: Domain
Simply put, a field is a set of additive commutative groups and multiplicative commutative groups. Addition and multiplication are related by the law of distribution. Generally speaking, when a set is a domain, write this set as F (the first letter of the domain).
Because it is an additive commutative group, there are addition and subtraction in the field, and 0 exists.
Because it is a multiplicative commutative group, there are multiplication and division in the domain, and there is 1.
In this way, the fields with 0 and 1 contain all rational numbers (all integers are generated by addition and then divided), just as we calculated above.
Domain is a perfect concept. Common domains include rational number set q (minimum domain! ), real number set r, complex number set c, etc. It is also called rational number field, real number field and complex number field in universities.
Incidentally, the sources of symbols such as rational number set Q, real number set R and complex number set C are also English initials. There are many similar signs in mathematics, such as set S (set). You can find out for yourself why they are all marked like this, which is very interesting.
The study of abstract algebra begins with groups, rings, modules and fields, but it is generally not studied except for students in mathematics department. "Algebra" in middle school is "replacing numbers with letters", commonly known as elementary algebra; In the university, it means "the nature of abstract axiomatic system", commonly known as modern algebra or abstract algebra.