This is a list of definitions and theorems concerning field extensions, that I’m compiling for my learning. Throughout, let 
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Category: Math
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On Renaming Universal Algebra
Especially in comparatively recent times, researchers in the subject of universal algebra have sought to replace that name, leading people to choose an alternative term instead. In this post, I discuss my thoughts on this matter.
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Algebraic Structures Applicable to All Sets
This exploration is inspired by the fact that, given the Axiom of Choice, any set can be given a group structure. (Fine print: throughout the rest of this post, we assume all sets are nonempty.) In fact, these two statements are equivalent. Questions about similar statements have popped up in other contexts; for example, in Topology as an Algebraic Structure, the question arises as to whether any infinite set can be given a field structure. (Clearly, not every finite set can be given a field structure, since we have the classification of finite fields.)
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Construction of the Reals from Decimals
In typical real analysis classes, the presentation of real numbers that is generally given in schools (starting from rationals) is considered insufficiently rigorous, and replaced by a construction involving Dedekind cuts. However, in this post, we investigate a formalization that derives directly from the presentation given in schools — specifically, using decimals — and study its equivalence to the Dedekind cut formulation.
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Characteristic of a Monoid
In ring theory, the characteristic is defined as the min 
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Archimedes and the Area of a Circle
In this post, we claim that the method of exhaustion developed by Archimedes to calculate the area of a circle is in fact sufficiently rigorous.
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Sine Angle Addition Functional Equation 2
In this post, we continue the discussion from this post, where we investigate the sine angle addition functional equation
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Normal Subset of a Group
We attempt to generalize the notion of “normal subgroup,” used in studying kernels and homomorphisms, to a “normal subset” that does not have the requirement of being a subgroup, and we see whether it is possible for a normal subset to exist that is not a normal subgroup.
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On the Philosophical Foundations of ZFC
In this post, we discuss some questions concerning the philosophical foundations of ZFC.
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On the Definition of Mathematical Rigor
My philosophical post on the circularity of mathematics implies that formal logic must rest ultimately on an informal foundation. In this post, I discuss what that means for our definition of mathematical rigor, which will be an important reference for future philosophical discussions.
