Field Extensions - Remarks by Lovett - Page 326 .... ....

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In summary, Lovett's remarks explain that in the quotient ring ##K##, the equation ##a(x)q(x)+b(x)p(x)=1## implies that ##\overline{ a(x) q(x) } = 1##. This is because when substituting ##\alpha## for ##x##, the equation collapses to ##a( \alpha ) q( \alpha ) = 1## due to the fact that ##p(\alpha)=0##.
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I am reading "Abstract Algebra: Structures and Applications" by Stephen Lovett ...

I am currently focused on Chapter 7: Field Extensions ... ...

I need help with some remarks of Lovett following Theorem 7.1.12 and Example 7.1.13 on page 326 ...The remarks by Lovett read as follows:

In the above remarks from Lovett, we read the following:

" ... ... In the quotient ring ##K##, this implies that ##\overline{ a(x) q(x) } = 1##. Thus in ##K, \ a( \alpha ) q( \alpha ) = 1##. ... ... "My question is as follows:

Can someone please explain exactly why/how it is that ##\overline{ a(x) q(x) } = 1## implies that ##a( \alpha ) q( \alpha ) = 1## ... ... ?Help will be appreciated ...

Peter

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In the equation ##a(x)q(x)+b(x)p(x)=1##, substitute ##\alpha## for ##x##. Since ##p(\alpha)=0## (we were told ##\alpha## is a root of ##p(x)##) the equation collapses to the desired result.

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What are field extensions and why are they important?

Field extensions are mathematical structures that extend a given mathematical field by adding new elements to it. They are important because they allow us to study new objects and concepts that were not previously accessible in the original field.

How do field extensions relate to algebraic equations?

Field extensions are closely related to algebraic equations because they can be used to find solutions to certain types of equations. In particular, field extensions can be used to find roots of polynomial equations over a given field.

What is the difference between a finite and infinite field extension?

A finite field extension is one in which the added elements are a finite number, while an infinite field extension has an infinite number of added elements. In other words, a finite extension has a finite degree (number of elements) while an infinite extension has an infinite degree.

How are field extensions related to Galois theory?

Field extensions are a central concept in Galois theory, which is a branch of abstract algebra that studies symmetries and structures of fields. Field extensions allow us to study the symmetries and structures of larger fields by understanding their relationship to smaller fields.

What are some applications of field extensions in real-world problems?

Field extensions have many practical applications, such as in cryptography, coding theory, and computer science. They are also used in physics and engineering to model and solve problems related to complex systems and structures.

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