I am reading Kenneth Ireland and Michael Rosen's book, "A Classical Introduction to Modern Number Theory" ... ...(adsbygoogle = window.adsbygoogle || []).push({});

I am currently focused on Chapter 12: Algebraic Number Theory ... ...

I need some help in order to follow a basic result in Section 1: Algebraic Preliminaries ...

The start of Section 1 reads as follows:

QUESTION 1

In the above text by Ireland and Rosen, we read the following:

"... ... Suppose ##\alpha_1, \alpha_2, \ ... \ ... \ , \alpha_n## is a basis for ##L/K## and ##\alpha \in L##.

Then ##\alpha \alpha_i = \sum_j a_{ ij } \alpha_j## with ##a_{ ij } \in K## ... ... "

"My question is ... ... how do Ireland and Rosen get ##\alpha \alpha_i = \sum_j a_{ ij } \alpha_j ## ... ... ?

My thoughts are as follows ...

Given ##L/K##, we have that ##L## is a vector space over ##K##.

... we then let ##\alpha_1, \alpha_2, \ ... \ ... \ , \alpha_n## be a basis for ##L## as a vector space over ##K##

( i take it that that is what I&R mean by "... ... Suppose ##\alpha_1, \alpha_2, \ ... \ ... \ , \alpha_n## is a basis for ##L/K##")

... we then let ##\alpha \in L## ... ... then there exist ##a_1, a_2, \ ... \ ... \ , a_n \in K##

such that

##\alpha = a_1 \alpha_1 + a_2 \alpha_2 + \ ... \ ... \ a_n \alpha_n##

so that

##\alpha \alpha_i = ( a_1 \alpha_1 + a_2 \alpha_2 + \ ... \ ... \ a_n \alpha_n ) \alpha_i ## ... ... ... (1)

... BUT ...

Ireland and Rosen write (see above)

##\alpha \alpha_i = \sum_j a_{ ij } \alpha_j##

##= a_{ i1 } \alpha_1 + a_{ i2 } \alpha_2 + \ ... \ ... \ + a_{ in } \alpha_n## ... ... ... (2)

My question is ... how do we get expression (1) equal to (2) ... ...

QUESTION 2

In the above text by Ireland and Rosen, we read the following:

"... ...The norm of ##\alpha, N_{ L/K } ( \alpha )## is ##\text{ det} (a_{ ij }) ## ... ...

I cannot fully understand the process involved in forming the norm ... can someone please explain ... preferably via a simple example ...

Hope someone can help ...

Peter

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# I Basics of Field Extensions ... ... Ireland and Rosen, Ch 12

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