Mathematics Asked by Billy Rubina on November 25, 2020

I am trying to understand the solution of the following exercise:

A number $a$ is unipotent if $aneq1$ and $a^2equiv 1 pmod{p}$.

- Show that when $p$ is prime, the only unipotent class is $p-1$.

The answer is:

- $(p-1)xequiv1 pmod{p}$ has a unique solution in $Bbb{Z}/pBbb{Z}$. This solution is $xequiv -1 pmod{p}$. It is the same as $x=p-1$.

I may be doing some very silly mistake: We want to prove that $a=p-1$ is the only number such that $a^2equiv 1 pmod{p}$ but why does the above proof proves it? Aren’t we proving only that $a$ is unipotent without – somehow – checking it for $(p-1),(p-2),(p-3),dots$?

Frankly I'm not quite sure what the answer's proof is trying to say, but the easiest way to see this result is to note that $Bbb Z/pBbb Z$ is a field, and thus the polynomial $x^2-1$ can only have at most two roots (i.e., as many roots as its degree). As $x=pm1$ both solve this polynomial, these are the only roots. As we're excluding $x=1$, this shows that there is a unique unipotent element modulo $p$.

That being said, in the case $p=2$ there are actually no unipotent elements since $1=-1mod 2$ is the only element that squares to $1$ in $Bbb Z/2Bbb Z$.

Correct answer by shibai on November 25, 2020

2 Asked on November 6, 2021 by navyism

1 Asked on November 6, 2021

adjoint operators banach spaces functional analysis operator theory

0 Asked on November 6, 2021 by marcelo-rm

2 Asked on November 6, 2021 by exodd

examples counterexamples trigonometric series trigonometry upper lower bounds

1 Asked on November 6, 2021

calculus convergence divergence power series sequences and series

1 Asked on November 6, 2021 by plasmacel

algebraic geometry euclidean geometry geometry linear algebra vectors

3 Asked on November 6, 2021 by jacobcheverie

elementary set theory functions proof verification proof writing

1 Asked on November 6, 2021

3 Asked on November 6, 2021 by domonoxyledyl

2 Asked on November 6, 2021 by josef-hlava

2 Asked on November 6, 2021 by dhrubajyoti-bhattacharjee

2 Asked on November 6, 2021 by aladin

4 Asked on November 6, 2021

5 Asked on November 6, 2021

3 Asked on November 6, 2021 by charlie-chang

1 Asked on November 6, 2021

algebraic geometry commutative algebra derived functors schemes sheaf cohomology

1 Asked on November 6, 2021 by pythusiast

Get help from others!

Recent Answers

- Joshua Engel on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- haakon.io on Why fry rice before boiling?

Recent Questions

- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?
- Does Google Analytics track 404 page responses as valid page views?

© 2022 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP, SolveDir