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I am reading David S. Dummit and Richard M. Foote : Abstract Algebra ...
I am trying to understand the example on Finite Fields in Section 13.5 Separable and Inseparable Extensions ...The example reads as follows:
My questions are as follows:
Question 1In the above text from D&F we read the following:
" ... ... If ##\mathbb{F}## is of dimension ##n## over its prime subfield ##\mathbb{F}_p##, then ##\mathbb{F}## has precisely ##p^n## elements. ... ... "Can someone please explain why, exactly, this follows?
Question 2
In the above text from D&F we read the following:
" ... ... Since the multiplicative group ##\mathbb{F}^{ \times }## is (in fact cyclic) of order ##p^n - 1##, we have ##\alpha^{ p^n - 1 } = 1## for every ##\alpha \ne 0 in \mathbb{F}## ... ... "Can someone give me the exact reasoning concerning why ##\mathbb{F}^{ \times }## being of order ##p^n - 1## implies that ##\alpha^{ p^n - 1} = 1## for every ##\alpha \ne 0## in ##\mathbb{F}## ... ... ?(I am guessing that for some reason I cannot explain, that ##\mathbb{F}^{ \times }## being of order ##p^n - 1 ## implies that the characteristic is ##p^n - 1## ... ... but why does it mean this is the case ...? )
Hope someone can help ...
Peter
I am trying to understand the example on Finite Fields in Section 13.5 Separable and Inseparable Extensions ...The example reads as follows:
My questions are as follows:
Question 1In the above text from D&F we read the following:
" ... ... If ##\mathbb{F}## is of dimension ##n## over its prime subfield ##\mathbb{F}_p##, then ##\mathbb{F}## has precisely ##p^n## elements. ... ... "Can someone please explain why, exactly, this follows?
Question 2
In the above text from D&F we read the following:
" ... ... Since the multiplicative group ##\mathbb{F}^{ \times }## is (in fact cyclic) of order ##p^n - 1##, we have ##\alpha^{ p^n - 1 } = 1## for every ##\alpha \ne 0 in \mathbb{F}## ... ... "Can someone give me the exact reasoning concerning why ##\mathbb{F}^{ \times }## being of order ##p^n - 1## implies that ##\alpha^{ p^n - 1} = 1## for every ##\alpha \ne 0## in ##\mathbb{F}## ... ... ?(I am guessing that for some reason I cannot explain, that ##\mathbb{F}^{ \times }## being of order ##p^n - 1 ## implies that the characteristic is ##p^n - 1## ... ... but why does it mean this is the case ...? )
Hope someone can help ...
Peter