1
vote
1answer
59 views

Proof that $a\equiv b \pmod n \iff a \pmod n = b\pmod n$

Proof that for every $a,b \in \mathbb Z,\ n \in \mathbb N$, that $$a\equiv b \pmod n \iff a \pmod n = b \pmod n.$$ My approach is: $n\mid a$ and $n\mid b$ $a\equiv b \pmod n \iff \exists x,y: ...
0
votes
1answer
33 views

How solve $[20]_3^{-1}$?

What does this mean, $[20]_3^{-1}$? it's from the topic rings, fields and residue classes. Can you give me a hint how to solve this?
-1
votes
1answer
80 views

What is zero times zero

What is zero zeros? What are no nothings? From a mathematical point of view it would be my thing, but none of us are educated that much in math, so I am curious to hear an expert opinion.
0
votes
1answer
32 views

Proving that operations give equal results given equal inputs

I was reading about the 9 or 12 basic properties of 'fields' (if that's what they're called) in a book called Spivak's Calculus, 3rd Edition, and got quite befuddled by dealing with as basic stuff as ...
1
vote
2answers
53 views

field number theory question

If we have ${a+b\sqrt{-1}}$ for a,b in ${Z_p}$, with $p$ as an odd prime, with $\sqrt{-1}^2=-1$, how do we show that $a+b\sqrt{-1}$ has a multiplicative inverse iff $a-b\sqrt{-1}$ has a multiplicative ...
1
vote
0answers
78 views

A Gauss sum over a field.

Let $K$ be a field (not necessarily $\mathbb C$) and let $\zeta=\zeta_n$ be a primitive $n$th root of unity in $\bar K$. I would like to know if there is a formula calculating $$ \sum_{k=1}^n ...
3
votes
1answer
101 views

Why don't I end up with the same splitting field?

I've understood that the splitting field of $x^4+2$ and the splitting field of $x^4-2$ over $\mathbb{Q}$ are both the field $\mathbb{Q}(\sqrt[4]{2} , i)$. With degree $8$ over $\mathbb{Q}$. This ...
0
votes
3answers
146 views

Weird Sub-ring/field? question

Let $R=\{\frac{n}{10^{k}} \mid (n,k) \in Z, k>-1\}$ which is the sub-ring of the Rational numbers ( assumed true) Consider S a subset of R $S= \{(3/10),(33/100),(333/100),...\}$ Show that this ...
3
votes
2answers
115 views

Lost on algebra notation

I'm in a basic number theory course and, never having taken college algebra, I'm lost on some of the notation. I'm wondering what some of these notations mean: 1: $\mathbb{Q}(i)$ ... I know that what ...
0
votes
1answer
642 views

Cyclotomic polynomial over a finite prime field [duplicate]

Possible Duplicate: Irreducible factors of $X^p-1$ in $(\mathbb{Z}/q \mathbb{Z})\[X\]$ Let $p$ be a prime number. Let $l$ be an odd prime number such that $l \neq p$. Let $X^l - 1 \in ...
4
votes
4answers
4k views

Finding inverse of polynomial in a field

I'm having trouble with the procedure to find an inverse of a polynomial in a field. For example, take: In $\frac{\mathbb{Z}_3[x]}{m(x)}$, where $m(x) = x^3 + 2x +1$, find the inverse of $x^2 + ...