I Prime Subfiellds - Lovett, Proposition 7.1.3 ...

1. Apr 14, 2017

Math Amateur

I am reading Abstract Algebra: Structures and Applications" by Stephen Lovett ...

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

I need help with the proof of, or at least some remarks concerning, Proposition 7.1.3 ...

Proposition 7.1.3 plus some introductory remarks (proof?) reads as follows:

In the above text from Lovett we read the following:

"... ... However, the multiplication on these elements as defined by distributivity gives this set of elements the structure of $\mathbb{F}_p = \mathbb{Z} / p \mathbb{Z}$. ... ... "

... ... BUT ... the subfield contains elements $0, 1, 2, 3, 4, 5, \ ... \ ... \ (p -1)$

... and being a field, it contains divisions of these elements such as $1/2, 3/5 \ ... \ ... \ ...$

... so how can this subfield be equal to $\mathbb{Z} / p \mathbb{Z}$ ... ... ?

Hope someone can help ...

Peter

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2. Apr 14, 2017

andrewkirk

The missing piece is that a field's characteristic must be either zero or a prime number, otherwise there will be divisors of zero (can you see why?), which would disqualify $F$ from being a field. We then use the theorem that $\mathbb Z_p$ is a field for $p$ prime, so that the fractions you listed are all modular integers in $\mathbb Z_p$. For instance $1/2=3$ in $\mathbb Z_5$ and $3/5=2$ in $\mathbb Z_7$.

3. Apr 14, 2017

Math Amateur

Thanks for the help Andrew ...

Just reflecting on what you have said ...

However, can you give me an indication of how exactly we can demonstrate that $1/2 = 3$ in $\mathbb{Z}_5$ and $3/5 = 2$ in $\mathbb{Z}_7$ ...

Then I will get an idea of what is going on ...

Peter

4. Apr 14, 2017

andrewkirk

For the first one, $3\times 2=6 = 5+1$, so that $3_5\times 2_5=1_5$ where the 5 subscript denotes 'mod 5' (technically speaking, $k_p$ denotes $k+p\mathbb Z$).

Similarly, $2\times 5=10=7+3$, so that $2_7\times 5_7=3_7$.

A good way of thinking of it is as a clock, with the numbers 0 to p-1 marked around the outside. and a single hand. Asking 'what is 1 / 2 mod 5' is the same as asking what rotation (past how many marked numbers) must I perform 2 times, starting at 0, to end up at 1?

5. Apr 14, 2017

Math Amateur

Thanks Andrew ... really appreciate your help ..

Peter