Field Theory: Nicholson, 6.2 Algebraic Extensions - Example 14 (p. 282) Solution

In summary, field theory is a branch of mathematics that studies algebraic structures called fields, which are sets of elements that can be added, subtracted, multiplied, and divided according to certain rules and axioms. Nicholson, the author of the book "Introduction to Abstract Algebra," is mentioned in this example as he is the source of the concept being discussed. Algebraic extensions are larger fields that contain all the elements of a smaller field and additional elements that are solutions to certain polynomial equations. Example 14 on page 282 is about finding the minimal polynomial for an algebraic extension and illustrates how to find the polynomial that satisfies the square root of 2 in the field of rational numbers. Finding minimal polynomials is important in field theory as it
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I am reading Nicholson: Introduction to Abstract Algebra, Section 6.2 Algebraic Extensions.

Example 14 on page 282 (see attachment) reads as follows:

-------------------------------------------------------------------------------------------

Example 14. Let [TEX] E \supseteq F [/TEX] be fields and let [TEX] u, v \in E [/TEX].

If u and u + v are algebraic over F, show that v is algebraic over F.

-------------------------------------------------------------------------------------------

In the solution, Nicholson writes the following:

Solution. Write L = F(u + v) so that L(u) = F(u, v). ... ... etc etc
Can someone please show me (formally and exactly) why [TEX] L = F(u + v) \Longrightarrow L(u) = F(u, v) [/TEX].

Peter

[This has also been posted on MHF]
 
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Peter said:
I am reading Nicholson: Introduction to Abstract Algebra, Section 6.2 Algebraic Extensions.

Example 14 on page 282 (see attachment) reads as follows:

-------------------------------------------------------------------------------------------

Example 14. Let [TEX] E \supseteq F [/TEX] be fields and let [TEX] u, v \in E [/TEX].

If u and u + v are algebraic over F, show that v is algebraic over F.

-------------------------------------------------------------------------------------------

In the solution, Nicholson writes the following:

Solution. Write L = F(u + v) so that L(u) = F(u, v). ... ... etc etc
Can someone please show me (formally and exactly) why [TEX] L = F(u + v) \Longrightarrow L(u) = F(u, v) [/TEX].

Peter

[This has also been posted on MHF]
We have $L=F(u+v)$. Thus $L(u)=(F(u+v))(u)=F(u+v,u)$.

We claim that $F(u+v,u)=F(v,u)$. Now that $u+v,u\in F(v,u)$. Thus $F(u+v,u)\subseteq F(v,u)$. Also, $u,v\in F(u+v,u)$. This is simply because $u+v-u$ is in $F(u+v,u)$. So we have $F(v,u)\subseteq F(u+v,u)$.
Thus we have $F(u+v,u)=F(v,u)$.

From here it's easy to get $L=F(v,u)$.
 

Related to Field Theory: Nicholson, 6.2 Algebraic Extensions - Example 14 (p. 282) Solution

1. What is field theory?

Field theory is a branch of mathematics that studies algebraic structures called fields. Fields are sets of elements that can be added, subtracted, multiplied, and divided, satisfying certain rules and axioms.

2. Who is Nicholson and why is he mentioned in this example?

Nicholson refers to Peter Nicholson, a mathematician who wrote the book "Introduction to Abstract Algebra" where this example is taken from. He is mentioned because he is the author of the text being used to illustrate the concept of field theory.

3. What are algebraic extensions?

An algebraic extension is a field extension where every element of the extension is a root of some polynomial with coefficients in the base field. In simpler terms, it is a larger field that contains all the elements of the smaller field and additional elements that are solutions to certain polynomial equations.

4. What is Example 14 on page 282 about?

Example 14 is about finding the minimal polynomial for an algebraic extension. In this case, the given extension is the field of rational numbers and the additional element is the square root of 2. The solution shows how to find the polynomial that the square root of 2 satisfies, which is the minimal polynomial for this algebraic extension.

5. Why is finding minimal polynomials important in field theory?

Finding minimal polynomials is important because they provide a way to represent elements of larger fields using elements of smaller fields. It also helps in understanding the structure and properties of algebraic extensions, which are essential in many areas of mathematics and science.

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