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Web of analogies between integers/rationals and finite fields: $$\mathbb{z} \iff \mathbb{F}_{q}[T]$$ $$\mathbb{z}[X] \iff \mathbb{F}_{q}[T,X]$$

Here $q = p^t$, where $p$ is an odd prime. I am not able to understand what is analogy here ?

Reference : http://www.math.rutgers.edu/~sk1233/courses/ANT-F14/lec10.pdf

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One of the important analogies is that the ring of integers of the number field $\mathbb{Q}$ is a PID, namely $\mathbb{Z}$, and that the ring of integers of the global field $\mathbb{F}_p(T)$ is again a PID, namely $\mathbb{F}_p[T]$. This is quite important in algebraic number theory. A number field $K$ has characteristic zero, whereas a function field, which is a finite extension of $\mathbb{F}_p(T)$ has characteristic $p>0$. Now many arguments holding for number fields can be applied to function fields, because $\mathbb{F}_p[T]$ is also a PID.

Note that the context of number theory seems appropriate, since the lecture you have linked is called "Number Theory (Fall 2014)".

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    what is $\mathbb{F}_{p}(T)$ ? – 2017-02-17
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    $\mathbb{F}_p(T)$ is the field of fractions of $\mathbb{F}_p[T]$, i.e., the field of rational functions with coefficients in $\mathbb F_p$. The analogy is that $\mathbb{Q}$ is the field of fractions of $\mathbb{Z}$. – 2017-02-17