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Simple Enlargement Body 📂Abstract Algebra

Simple Enlargement Body

Definition 1

If an extension field EE of FF satisfies E=F(α)E = F( \alpha ) for some αE\alpha \in E, then EE is called a Simple Extension of FF.

Explanation

Simply put, F(α)F ( \alpha ) can be seen as an expansion by adding just one α\alpha that was not in FF. Speaking in terms of the field of real numbers R\mathbb{R}, adding iCi \in \mathbb{C} to its extension field C\mathbb{C} results in R(i)=C\mathbb{R} ( i ) = \mathbb{C}.

An important fact is that for αE\alpha \in E, if E=F(α)E = F ( \alpha ), then all βE\beta \in E are β=b0+b1α++bnαn \beta = b_{0} + b_{1} \alpha + \cdots + b_{n} \alpha^n uniquely represented like this. In this case, {bk}k=1n\left\{ b_{k} \right\}_{k =1}^{n} is an element of FF, and thinking about the field of complex numbers as a simple extension of real numbers, it’s easy to see that all complex numbers zCz \in \mathbb{C} can be represented for some x,yRx , y \in \mathbb{R} as z=x0+y0i+x1i2+y1i3+=x+iy z = x_{0} + y_{0} i + x_{1} i^2 + y_{1} i^3 + \cdots = x + i y

On another note, an interesting example of simple extensions includes Gaussian integers ii and ω\omega added to integers, such as Gaussian integers Z[i]\mathbb{Z} [i] and Eisenstein integers Z[ω]\mathbb{Z} [\omega].


  1. Fraleigh. (2003). A first course in abstract algebra(7th Edition): p270. ↩︎