# quadratic again.. may be tricky

• November 28th 2007, 09:58 AM
joanne_q
hi, i think this one is a bit trickier.

for an odd prime number "p", let $F_p$ be the field with "p" elements. i.e. the integers {0,....p-1} with addition and multiplication defined modulo "p".
So how many quadratic forms are there on the vector space $F^n_p$ and why?

• November 28th 2007, 11:35 AM
ThePerfectHacker
Quote:

Originally Posted by joanne_q
So how many quadratic forms are there on the vector space $F^n_p$ and why?

• November 28th 2007, 12:03 PM
joanne_q
thats the problem i have myself.
i dont quite understand the question so im trying to see if any of you guys have an idea of what i am meant to be doing here :)
• November 28th 2007, 12:04 PM
joanne_q
here is the question as on the paper:

for an odd prime number "p", let $F_p$ be the field with "p" elements. i.e. the integers {0,....p-1} with addition and multiplication defined modulo "p", how many quadratic forms are there on the vector space $F^n_p$ and why?
• November 28th 2007, 12:06 PM
joanne_q
a quadratic form by the way would be the quadratic equation corresponding to the symmetric matrix :

so for example, a 2x2 matrix would produce the follow quadratic form:
ax² + 2hxy + by² = 1.

i hope that makes it a bit more clearer..

any ideas on how to solve the original question then?
• November 28th 2007, 06:28 PM
ThePerfectHacker
Searching the internet reveals that a quadradic form is a symettric quadradic polynomial, for example, $x^2+y^2+4xy$. Thus we you are talking about a vector space $\mathbb{F}_p^n$ it seems to me are you trying to count all quadric forms of $x_1,...,x_n$ variables. So it has the form: $a_1x_1^2+...+a_nx_n^2+b_1x_1x_2+...$. Now just do a counting argument. Count all such coefficients. The $a_i$ for each $1\leq i \leq n$ can be one of $0,1,...,p-1$, thus in total we have $p^n$ combinations. Now the $b_i$ are all combinations of $x_i,x_j$ with $i\not = j$ there are $C_{n,2}=n(n-1)/2$ such possibilities. For each coefficient we have $p$ possibilities that we have $p^{n(n-1)/2}$. Altogether we have $p^{n(n-1)/2+n)}=p^{n(n+1)/2}$. But in counting this we have included that zero polynomial which is not a quadradic form so we just subtract $1$ from that.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 21, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9669026732444763, "perplexity": 407.00732479280566}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207927388.78/warc/CC-MAIN-20150521113207-00254-ip-10-180-206-219.ec2.internal.warc.gz"}