Field:F3: Difference between revisions
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This field, denoted <math>F_3</math> or <math>GF(3)</math>, is the unique field of three elements. It can be defined as the ring of integers modulo <math>3</math>. | This field, denoted <math>F_3</math> or <math>GF(3)</math>, is the unique field of three elements. It can be defined as the ring of integers modulo <math>3</math>. | ||
== | ==Related groups== | ||
{| class="wikitable" border="1" | |||
! Group functor !! Value !! GAP ID | |||
|- | |||
| additive group || [[cyclic group:Z3]] || (3,1) | |||
|- | |||
| multiplicative group || [[cyclic group:Z2]] || (2,1) | |||
|- | |||
| general affine group of degree one || [[symmetric group:S3]] || (6,1) | |||
|- | |||
| [[general linear group of degree two]] || [[general linear group:GL(2,3)]] || (48,29) | |||
|- | |||
| [[special linear group of degree two]] || [[special linear group:SL(2,3)]] || (24,3) | |||
|- | |||
| [[projective general linear group of degree two]] || [[symmetric group:S4]] || (24,12) | |||
|- | |||
| [[projective special linear group of degree two]] || [[alternating group:A4]] || (12,3) | |||
|- | |||
| [[projective special linear group]] of degree three || [[projective special linear group:PSL(3,3)]] || order 5616, no GAP ID. | |||
|- | |||
| upper-triangular unipotent matrix group of degree three || [[prime-cube order group:U(3,3)]] || (27,3) | |||
|} | |||
==GAP implementation== | |||
{{GAP finite field of prime order|3}} | |||
Latest revision as of 11:00, 9 December 2023
This article is about a particular field, i.e., a field unique up to isomorphism. View a complete list of particular fields
Definition
This field, denoted or , is the unique field of three elements. It can be defined as the ring of integers modulo .
Related groups
| Group functor | Value | GAP ID |
|---|---|---|
| additive group | cyclic group:Z3 | (3,1) |
| multiplicative group | cyclic group:Z2 | (2,1) |
| general affine group of degree one | symmetric group:S3 | (6,1) |
| general linear group of degree two | general linear group:GL(2,3) | (48,29) |
| special linear group of degree two | special linear group:SL(2,3) | (24,3) |
| projective general linear group of degree two | symmetric group:S4 | (24,12) |
| projective special linear group of degree two | alternating group:A4 | (12,3) |
| projective special linear group of degree three | projective special linear group:PSL(3,3) | order 5616, no GAP ID. |
| upper-triangular unipotent matrix group of degree three | prime-cube order group:U(3,3) | (27,3) |
GAP implementation
The field can be defined using GAP's GF function:
GF(3)
It can also be defined using the ZmodnZ function:
ZmodnZ(3)