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 IAmABread June 5th, 2017 06:14 AM

Infinite field with finite characteristic

I need a good example of infinite field with finite characteristic. Can anyone help?

 Maschke June 5th, 2017 06:19 AM

Quote:
 Originally Posted by IAmABread (Post 571993) I need a good example of infinite field with finite characteristic. Can anyone help?
Algebraic closure of any finite field.

 ABVictor June 5th, 2017 07:21 AM

Rational functions over a finite field.

 IAmABread June 5th, 2017 08:01 AM

I would like you to be more precisive, if that is not a problem. Can you clarify it using a particular field? It will surely become clearer to me.

 ABVictor June 5th, 2017 08:39 AM

The field of all fractions $\displaystyle \frac{f(x)}{g(x)}$ where $\displaystyle f(x),g(x)\in \mathbb{F}_7[x],\ g(x)\neq 0.$

It is the field of rational functions in $\displaystyle x$
$\displaystyle \frac{a_n x^n + a_{n-1} x^{n-1} +\ \ldots\ + a_0}{b_mx^m+b_{m-1}x^{m-1}+\ \ldots\ +b_0}$
with coefficients $\displaystyle a_i, b_j$ in $\displaystyle \mathbb{F}_7.$

 Maschke June 5th, 2017 09:50 AM

* No finite field is algebraically closed. https://math.stackexchange.com/quest...d-fields-exist

* Every field has an algebraic closure. https://en.wikipedia.org/wiki/Algebraic_closure

Therefore the algebraic closure of, say, $\mathbf F_2$ is an infinite field of characteristic $2$.

 IAmABread June 5th, 2017 03:56 PM

Does $\mathbb{F}_7$ have a structure like $\mathbb{Z}_7$? Sorry for all those elementary questions, but I am right after first lecture of field theory, so I can't know much.

 Maschke June 5th, 2017 04:04 PM

Quote:
 Originally Posted by IAmABread (Post 572023) Does $\displaystyle \mathbb{F}_7$ have a structure like $\displaystyle \mathbb{Z}_7$?
Yes, $\mathbf F_p$, the field with $p$ elements ($p$ a prime) is the same as $\mathbb Z_p$, the ring of integers mod $p$.

Where it gets weird though is for the finite fields that are nontrivial powers of $p$. For example $\mathbb Z_4$ is not a field because $2 \times 2 = 0$. But there is a field of order $4$, whose addition and multiplication table you might try working out.

Quote:
 Originally Posted by IAmABread (Post 572023) Sorry for all those elementary questions, but I am right after first lecture of field theory, so I can't know much.
Best to not worry too much about these things at the moment. Your questions are good but the answers aren't elementary. Finite fields are tricky and so is the proof of the existence of algebraic closures.

 IAmABread June 8th, 2017 09:28 AM

Okay, let's say I take coefficients from $\mathbb{Z}_3$. What will multiplication identity look like here? Is it just $1 + 0*x + 0*x^2 + 0*x^3 +...$ or something more complex? If not, then I assume characteristic is equal 3 according to what I typed. Am I right in all of this?

 Maschke June 8th, 2017 10:34 AM

Quote:
 Originally Posted by IAmABread (Post 572230) Okay, let's say I take coefficients from $\mathbb{Z}_3$.
Coefficients of what?

Quote:
 Originally Posted by IAmABread (Post 572230) What will multiplication identity look like here?
Of what ring? You seem to have some hidden assumptions, or you're working in a ring you haven't told us about.

Quote:
 Originally Posted by IAmABread (Post 572230) Is it just $1 + 0*x + 0*x^2 + 0*x^3 +...$ or something more complex?
Is this a ring of formal power series? Where did that come from?

Quote:
 Originally Posted by IAmABread (Post 572230) If not, then I assume characteristic is equal 3 according to what I typed. Am I right in all of this?
No. As far as I can tell your post is missing any context that would make it sensible. Don't mean for that to come out as overly critical. But you lost me at "coefficients." Coefficients of what?

In any event I don't believe the ring of formal power series over a prime-order field is a field. It's an integral domain though.

https://math.stackexchange.com/quest...ntegral-domain

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