Image of an integer polynomial in the integers mod 7
$begingroup$
Given p(x,y) = x2 - 7y2 - 24 ∈ ℤ[x,y], does the image in ℤ7[x,y] become x2 - 3 or x2 + 4? Or could I use either to determine whether or not p(x,y) has any solutions in ℤ[x,y]?
abstract-algebra polynomials modular-arithmetic
asked Dec 10 '18 at 18:50
Jon D.Jon D.
164
$endgroup$
add a comment |
$begingroup$
Given p(x,y) = x2 - 7y2 - 24 ∈ ℤ[x,y], does the image in ℤ7[x,y] become x2 - 3 or x2 + 4? Or could I use either to determine whether or not p(x,y) has any solutions in ℤ[x,y]?
abstract-algebra polynomials modular-arithmetic
asked Dec 10 '18 at 18:50
Jon D.Jon D.
164
$endgroup$
3
$begingroup$
Either will do, they are equivalent $pmod 7$.
$endgroup$
– lulu
Dec 10 '18 at 18:51
$begingroup$
Indeed, @lulu, I would have said that as polynomials over the field with seven elements, they are equal.
$endgroup$
– Lubin
Dec 11 '18 at 4:11
add a comment |
$begingroup$
Given p(x,y) = x2 - 7y2 - 24 ∈ ℤ[x,y], does the image in ℤ7[x,y] become x2 - 3 or x2 + 4? Or could I use either to determine whether or not p(x,y) has any solutions in ℤ[x,y]?
abstract-algebra polynomials modular-arithmetic
asked Dec 10 '18 at 18:50
Jon D.Jon D.
164
$endgroup$
Given p(x,y) = x2 - 7y2 - 24 ∈ ℤ[x,y], does the image in ℤ7[x,y] become x2 - 3 or x2 + 4? Or could I use either to determine whether or not p(x,y) has any solutions in ℤ[x,y]?
abstract-algebra polynomials modular-arithmetic
abstract-algebra polynomials modular-arithmetic
asked Dec 10 '18 at 18:50
Jon D.Jon D.
164
asked Dec 10 '18 at 18:50
Jon D.Jon D.
164
edited Dec 10 '18 at 18:51
Jon D.
asked Dec 10 '18 at 18:50
Jon D.Jon D.
164
asked Dec 10 '18 at 18:50
Jon D.Jon D.
164
asked Dec 10 '18 at 18:50
Jon D.Jon D.
164
164
3
$begingroup$
Either will do, they are equivalent $pmod 7$.
$endgroup$
– lulu
Dec 10 '18 at 18:51
$begingroup$
Indeed, @lulu, I would have said that as polynomials over the field with seven elements, they are equal.
$endgroup$
– Lubin
Dec 11 '18 at 4:11
add a comment |
3
$begingroup$
Either will do, they are equivalent $pmod 7$.
$endgroup$
– lulu
Dec 10 '18 at 18:51
$begingroup$
Indeed, @lulu, I would have said that as polynomials over the field with seven elements, they are equal.
$endgroup$
– Lubin
Dec 11 '18 at 4:11
3
3
$begingroup$
Either will do, they are equivalent $pmod 7$.
$endgroup$
– lulu
Dec 10 '18 at 18:51
$begingroup$
Either will do, they are equivalent $pmod 7$.
$endgroup$
– lulu
Dec 10 '18 at 18:51
$begingroup$
Indeed, @lulu, I would have said that as polynomials over the field with seven elements, they are equal.
$endgroup$
– Lubin
Dec 11 '18 at 4:11
$begingroup$
Indeed, @lulu, I would have said that as polynomials over the field with seven elements, they are equal.
$endgroup$
– Lubin
Dec 11 '18 at 4:11
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You could use either $x^2-3$ or $x^2+4$, or in fact $x^2+(4+n)times 7$ for any $ n in mathbb N;$
adding any multiple of $7$ doesn't change things modulo $7$.
answered Feb 4 at 1:03
J. W. TannerJ. W. Tanner
3,8871320
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3034329%2fimage-of-an-integer-polynomial-in-the-integers-mod-7%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You could use either $x^2-3$ or $x^2+4$, or in fact $x^2+(4+n)times 7$ for any $ n in mathbb N;$
adding any multiple of $7$ doesn't change things modulo $7$.
answered Feb 4 at 1:03
J. W. TannerJ. W. Tanner
3,8871320
$endgroup$
add a comment |
$begingroup$
You could use either $x^2-3$ or $x^2+4$, or in fact $x^2+(4+n)times 7$ for any $ n in mathbb N;$
adding any multiple of $7$ doesn't change things modulo $7$.
answered Feb 4 at 1:03
J. W. TannerJ. W. Tanner
3,8871320
$endgroup$
add a comment |
$begingroup$
You could use either $x^2-3$ or $x^2+4$, or in fact $x^2+(4+n)times 7$ for any $ n in mathbb N;$
adding any multiple of $7$ doesn't change things modulo $7$.
answered Feb 4 at 1:03
J. W. TannerJ. W. Tanner
3,8871320
$endgroup$
You could use either $x^2-3$ or $x^2+4$, or in fact $x^2+(4+n)times 7$ for any $ n in mathbb N;$
adding any multiple of $7$ doesn't change things modulo $7$.
answered Feb 4 at 1:03
J. W. TannerJ. W. Tanner
3,8871320
answered Feb 4 at 1:03
J. W. TannerJ. W. Tanner
3,8871320
answered Feb 4 at 1:03
J. W. TannerJ. W. Tanner
3,8871320
answered Feb 4 at 1:03
J. W. TannerJ. W. Tanner
3,8871320
3,8871320
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3034329%2fimage-of-an-integer-polynomial-in-the-integers-mod-7%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
3
$begingroup$
Either will do, they are equivalent $pmod 7$.
$endgroup$
– lulu
Dec 10 '18 at 18:51
$begingroup$
Indeed, @lulu, I would have said that as polynomials over the field with seven elements, they are equal.
$endgroup$
– Lubin
Dec 11 '18 at 4:11