Question

If $f(x)\in\Bbb Z_p [x]$ is irreducible with $p$ prime and $\deg ⁡f(x)=n$ then $\Bbb Z_p [x]/〈f(x)〉$ is a field with $p^n$ elements.

score 4.9k · Answer 1 · 2013-11-22 12:30:07

4.9k

Views

If $f(x)\in\Bbb Z_p [x]$ is irreducible with $p$ prime and $\deg ⁡f(x)=n$ then $\Bbb Z_p [x]/〈f(x)〉$ is a field with $p^n$ elements.

Published on 22 Nov 2013 - 12:30

score 501 · Answer 2 · 2013-11-23 19:50:30

501

Views

A sufficient condition for irreducibility of a polynomial in an extension field.

Published on 23 Nov 2013 - 19:50

score 1.6k · Answer 3 · 2013-11-23 23:13:17

1.6k

Views

Irreducible polynomial over a finite field?

Published on 23 Nov 2013 - 23:13

score 1.3k · Answer 4 · 2013-11-24 20:18:39

1.3k

Views

Irreducible factors for $x^q-x-a$ in $\mathbb{F}_p$.

Published on 24 Nov 2013 - 20:18

score 443 · Answer 5 · 2013-11-25 00:48:09

443

Views

Proof that a polynomial is irreducible for all $n \ne 4$.

Published on 25 Nov 2013 - 0:48

score 97 · Answer 6 · 2013-11-25 00:48:10

97

Views

Kronecker's Theorem - what can be deduced if $f$ is reducible?

Published on 25 Nov 2013 - 0:48

score 287 · Answer 7 · 2013-11-26 21:16:09

287

Views

Irreducibility of a particular polynomial

Published on 26 Nov 2013 - 21:16

score 154 · Answer 8 · 2013-11-29 17:39:50

154

Views

Prove that : $x^n+4$ is irreducible in $\mathbb{Z}[x]$.

Published on 29 Nov 2013 - 17:39

score 623 · Answer 9 · 2013-11-30 13:45:57

623

Views

Find the order of a polynomial

Published on 30 Nov 2013 - 13:45

score 204 · Answer 10 · 2013-11-30 16:03:34

204

Views

What is the smallest Finite Field in which the following polynomial is factorizable to irreducible factors?

Published on 30 Nov 2013 - 16:03

score 81 · Answer 11 · 2013-11-30 16:41:29

81

Views

Compute $a(x)b(x)+c(x)$ in $\mathrm{GF}(2^4)$ where the irreducible generator polynomial $x^4+x+1$.

Published on 30 Nov 2013 - 16:41

score 1.3k · Answer 12 · 2013-12-01 17:05:35

1.3k

Views

$x^{p-1} + ... + x^2 + x + 1$ is irreducible using Eisenstein's criterion?

Published on 01 Dec 2013 - 17:05

score 315 · Answer 13 · 2013-12-01 17:32:11

315

Views

$x^5 + 3x^2 - 7x - 1$ irreducible in $\mathbb{R}[x]$?

Published on 01 Dec 2013 - 17:32

score 498 · Answer 14 · 2013-12-02 04:11:48

498

Views

Irreducible polynomial in $\mathbb{C} [x,y ]$

Published on 02 Dec 2013 - 4:11

score 1.1k · Answer 15 · 2013-12-02 14:13:42

1.1k

Views

Different ways of proving a polynomial is irreducible in finite field.

Published on 02 Dec 2013 - 14:13

List Question

If $f(x)\in\Bbb Z_p [x]$ is irreducible with $p$ prime and $\deg ⁡f(x)=n$ then $\Bbb Z_p [x]/〈f(x)〉$ is a field with $p^n$ elements.

A sufficient condition for irreducibility of a polynomial in an extension field.

Irreducible polynomial over a finite field?

Irreducible factors for $x^q-x-a$ in $\mathbb{F}_p$.

Proof that a polynomial is irreducible for all $n \ne 4$.

Kronecker's Theorem - what can be deduced if $f$ is reducible?

Irreducibility of a particular polynomial

Prove that : $x^n+4$ is irreducible in $\mathbb{Z}[x]$.

Find the order of a polynomial

What is the smallest Finite Field in which the following polynomial is factorizable to irreducible factors?

Compute $a(x)b(x)+c(x)$ in $\mathrm{GF}(2^4)$ where the irreducible generator polynomial $x^4+x+1$.

$x^{p-1} + ... + x^2 + x + 1$ is irreducible using Eisenstein's criterion?

$x^5 + 3x^2 - 7x - 1$ irreducible in $\mathbb{R}[x]$?

Irreducible polynomial in $\mathbb{C} [x,y ]$

Different ways of proving a polynomial is irreducible in finite field.

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