Why 10-adics are not a field?

[aalireza]

I've got a question and I really need the answer! Why 10-adics are not a field? And generally, How can you be sure that a given set is a field or not? For example rational numbers are a field, but what about the others and how can you be sure?

 

[Landau]

You cannot decide whether a given set is a field. You can decide whether a given ring is a field. After all, a field is by definition a special kind of ring, namely a commutative ring in which every nonzero element is invertible. In particular, a field has no zero divisors (i.e. a field is a domain). If you can show that a certain ring has zero divisors, then it is not a field.

 

[aalireza]

@Landau:
My bad, you're right. But what is the zero divisors for 10-adics? Is there any example?

 

[Citan Uzuki]

See this: http://www.numericana.com/answer/p-adic.htm#composite-radix.

 

[blue_raver22]

I would like to clarify that the concept of 10-adics is a mathematical concept and not a scientific one. However, I can still provide an explanation as to why 10-adics are not a field.

Firstly, let's define what a field is. A field is a set of numbers where the operations of addition, subtraction, multiplication, and division are defined and follow certain rules. In a field, every non-zero number has a multiplicative inverse, meaning that it can be multiplied by another number to give a result of 1. Additionally, the operations of addition and multiplication are commutative and associative, and the distributive property also holds.

Now, 10-adics are a type of number system called the p-adic numbers, where p represents a prime number. In the case of 10-adics, the base number is 10. The 10-adic numbers are constructed by taking infinite sequences of digits and grouping them into numbers, similar to how we construct decimal numbers. However, unlike decimal numbers where the digits are multiplied by powers of 10, in 10-adics, the digits are multiplied by powers of 10 in the opposite direction (i.e. from right to left).

The main reason why 10-adics are not a field is that they do not follow the rule of multiplicative inverses. In other words, not every non-zero 10-adic number has a multiplicative inverse. For example, the number 2 in 10-adics does not have a multiplicative inverse since there is no 10-adic number that can be multiplied by 2 to give a result of 1. This violates one of the fundamental properties of a field.

To determine whether a given set is a field or not, one needs to check if it follows all the rules and properties of a field. For example, the set of rational numbers is a field because it follows all the rules of a field, including the existence of multiplicative inverses for every non-zero number. Other examples of fields include the set of real numbers, complex numbers, and finite fields.

In conclusion, 10-adics are not a field because they do not follow the fundamental properties of a field, specifically the existence of multiplicative inverses. To determine if a given set is a field, one needs to check if it follows all the rules and properties of a field.