# Tagged Questions

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### Is it possible to describe $Q(x)$ as the extension field of $R$ freely generated by $\{x\}$?

Given an integral domain $R$, the polynomial ring $R[x]$ can be defined as the commutative $R$-algebra freely generated by $\{x\}$. Also, let $Q$ denote the field of fractions associated to $R$. Then ...
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### Is there a universal property in this proposition (which regards field extensions)?

Since learning a bit of category theory, I am trying, as an exercise, to state results that I come across in categorical language. I am trying to do this with the following: Let ...
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### What is the left adjoint of the forgetful functor from fields to integral domains?

I quote from Wikipedia, regarding the construction of the field of fractions of an integral domain: "There is a categorical interpretation of this construction. Let $\mathcal{C}$ be the category ...
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### Why does Fld not have an initial object?

My Algebra book says that the category Fld of fields has no initial object. Why would $\{0,1\}$ not be an initial object? Does it not have a unique homomorphism to every other field?
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### Check that this is a category

Assume we have a fixed field $F$. We define objects as homomorphisms $\phi:F\rightarrow G$. Then we define morphisms between $\phi:F\rightarrow G$ and $\psi:F\rightarrow L$ as ring homomorphism from ...
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### How does one characterise the reals without points?

The Real number line is defined upto unique isomorphism in the category $Fld$ as a Dedekind complete ordered field. One can view the Reals geometrically with its usual topology. In pointless ...
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### A field extension $K \subseteq F$ is finitely-generated if and only if there exists a ring epimorphism $K[t_1 \cdots t_n] \twoheadrightarrow F$

Does this make sense as an alternative definition for a finitely-generated field extension?: A field extension $K \subseteq F$ is finitely-generated if and only if there exists a ring epimorphism ...
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### Should this be viewed as a serious issue with the meadow-theoretic approach?

Meadow theory (see here) allows us to apply the results and concepts of universal algebra to the study of fields. Obviously, this is very, very nice. However, I have the following issue with the ...
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### A field extension and equality up to isomorphisms

If we have in the category of rings two fields $F$ and $F'$ such that $F\hookrightarrow F'$ and $F'\hookrightarrow F$, do we have an isomorphism between $F$ and $F'$ ? If it is not always true, how ...
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### Field Extensions and their dimensions

There is a powerful theorem with respect to a field $F$ extensions and their dimensions. $F<E<K \ \Rightarrow [K:F] = [K:E][E:F]$ This is analogous to the famous Lagrange's theorem with ...
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### Rigidity of the category of fields

Let's call a category rigid if every self-equivalence is isomorphic to the identity. For example, $\mathsf{Set}$, $\mathsf{Grp}$, $\mathsf{Ab}$, $\mathsf{CRing}$ (MO/106838), $\mathsf{Top}$ ...
It is said that the category of fields $\mathsf{Fld}$ is ill-behaved, for example it is not an algebraic theory, does not have initial or terminal objects. In particular it is not presentable. On the ...