Does Every Field Have a Subfield Isomorphic to Q or Z mod p?

[ECmathstudent]

I just wrote an exam in Algebra II, and one of our true or false questions got me thinking. Does every field have a subfield which is isomorphic to the Q or Z mod p? I put it as true, for the wrong reasons, vaguely remembering a similar statement about integral domains and mixing it up. So, after the exam I was sure I was wrong.
But doesn't every field have the subfield {0,1}? Wouldn't this make the statement true, if somewhat vacuously?

 

[fourier jr]

it's called the prime subfield, which is the intersection of all subfields of the given field. it's isomorphic to Q or Z_p depending on whether the field has characteristic 0 or p

 

[ECmathstudent]

Oh, thanks. I don't have a book anymore, when I tried to look it up on google I wasn't really getting any results.

 

[Deveno]

yes.

it is quite easy to show this.

suppose 1+1+...+1 (n times) = 0.

then the subring generated by 1 is isomorphic to Zn.

however, since F is a field, none of the elements:

1, 1+1, 1+1+1, etc. can be a zero-divisor, which forces n to be prime.

on the other hand, suppose that n1 is never 0, for all n in Z+.

then the subring generated by 1 is isomorphic to the integers, and

any field containing the integers also contains the field of quotients of the integers, which is Q.

(also {0,1} is not a field, unless 1+1 = 0).

 

[blue_raver22]

I would approach this question from a mathematical perspective. First, let's define some terms. A field is a mathematical structure that satisfies certain properties, such as closure under addition, subtraction, multiplication, and division. For example, the set of rational numbers (Q) and the set of integers modulo a prime number (Z mod p) are both fields. A subfield is a subset of a field that is also a field, meaning it satisfies the same properties as the larger field.

Now, to answer the question, we need to consider the properties of fields and subfields. It is true that every field has a subfield isomorphic to Q or Z mod p. This is because every field contains a subset that is isomorphic (meaning structurally equivalent) to Q or Z mod p. This subset could be the entire field itself, or a smaller subset such as {0,1} in the case of Q.

However, it is important to note that not all subfields of a field are necessarily isomorphic to Q or Z mod p. For example, the field of real numbers (R) does not have a subfield isomorphic to Q, as Q is a proper subset of R. So while every field has the potential to contain a subfield isomorphic to Q or Z mod p, it is not a guarantee for all subfields.

In conclusion, the statement "every field has a subfield isomorphic to Q or Z mod p" is true, but it is important to understand the properties and limitations of fields and subfields in order to fully grasp the concept.