You're quoting the introduction of the article, which is notoriously a fuzzy abstract in all the wikipedia articles about mathematical concepts. Let's quote the actual (textual) formal definition (sec. 1.1) as it would stand in a textbook:
> Formally, a field is a set F together with two binary operations on F called addition and multiplication. A binary operation on F is a mapping F × F → F, that is, a correspondence that associates with each ordered pair of elements of F a uniquely determined element of F. The result of the addition of a and b is called the sum of a and b, and is denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, and is denoted ab or a ⋅ b. These operations are required to satisfy the following properties, referred to as field axioms. In these axioms, a, b, and c are arbitrary elements of the field F. [...]
Still, i don't know how to say it in another way but you probably don't have much experience in mathematics, even the first quote is arguably quite accurate.
> Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set?
Yes, unless told otherwise an operation is an internal binary operation, it's really the most common form. When it is not the output is notable enough to be specified.
> But if that was the case then "behave as the corresponding operations on rational and real numbers do" would mean that the fields would always need to be of infinite size wouldn't it? Because if the field had a limited number of elements and you added the last two (highest) elements together [...]
No, the important point here is your use of "highest". A set by default only has equality (and mappings, in and out) but no order relationship. So the most conservative interpretation of "behave as the corresponding operations on rational" would be to only include stuff that can be written using the 4 operations and equality, not ordering.
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Maybe i'm biaised by the fact that i know what a field is, but still, this particular intro is also how i would present a field: give the most common example and say which operations it has. It sure can create false intuitions like yours about the size, but this will always be the case when we use non-normalized language.
> Formally, a field is a set F together with two binary operations on F called addition and multiplication. A binary operation on F is a mapping F × F → F, that is, a correspondence that associates with each ordered pair of elements of F a uniquely determined element of F. The result of the addition of a and b is called the sum of a and b, and is denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, and is denoted ab or a ⋅ b. These operations are required to satisfy the following properties, referred to as field axioms. In these axioms, a, b, and c are arbitrary elements of the field F. [...]
Still, i don't know how to say it in another way but you probably don't have much experience in mathematics, even the first quote is arguably quite accurate.
> Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set?
Yes, unless told otherwise an operation is an internal binary operation, it's really the most common form. When it is not the output is notable enough to be specified.
> But if that was the case then "behave as the corresponding operations on rational and real numbers do" would mean that the fields would always need to be of infinite size wouldn't it? Because if the field had a limited number of elements and you added the last two (highest) elements together [...]
No, the important point here is your use of "highest". A set by default only has equality (and mappings, in and out) but no order relationship. So the most conservative interpretation of "behave as the corresponding operations on rational" would be to only include stuff that can be written using the 4 operations and equality, not ordering.
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Maybe i'm biaised by the fact that i know what a field is, but still, this particular intro is also how i would present a field: give the most common example and say which operations it has. It sure can create false intuitions like yours about the size, but this will always be the case when we use non-normalized language.