Infinite direct product of fields. Let $F$ be a field, and consider the infinite direct product$$F \times F \times F \times F \times \dots,$$i.e. $\prod_{i=0}^\infty F$, i.e. the direct product of a countable number of copies of $F$. The theory of bases for vector spaces shows that this direct product has a basis, and so is isomorphic to a direct sum of copies of $F$. How many copies of $F$ do I need $($i.e. what is the dimension of this direct product as an $F$-vector space$)$? Is it possible to exhibit this isomorphism between an infinite direct product and an infintie direct sum explicitly?
The dimension is the cardinality $|F|^{\aleph_0}$ according to the Theorem of Erdös-Kaplansky. See MO/49551 for a general formula of the dimension of an infinite family of vector spaces. All this belongs to "set-theoretic linear algebra", and no bases can be explicitly written down, since the axiom of choice plays a central role. Probably there are models of ZF where $\mathbb{Q} \times \mathbb{Q} \times \dotsc$ is no direct sum of copies of $\mathbb{Q}$; hopefully a set-theorist can add a reference for this.
To see the difference between infinite direct product and infinite direct sum (both involving countable number of copies) see their concrete realisations:
The vector space of all polynomials in $X$ over $F$, and the vector space of all formal power series (convergent or not) in $X$ over $F$. Now you can see what forms the basis in each case.