1
vote
2answers
52 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
72 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
95 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
0answers
55 views

Proof of Lempel-Golomb construction of Costas array

Can anyone please help me to prove Lempel-Golomb construction of Costas array, i.e., ${g_1}^i + {g_2}^j = 1$ forms costas array where $g_1$ and $g_2$ are primitive roots of a prime $p$ and $1\leq i ...
0
votes
3answers
142 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
109 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
565 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 ...
3
votes
4answers
3k 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 + ...