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sage.rings.finite_rings: Modularization fixes, # needs #36056

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merged 23 commits into from
Aug 13, 2023
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sage.rings.finite_rings: Modularization fixes, # needs #36056

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bfc5cdf
src/sage/rings/finite_rings/integer_mod.pyx: Move import from sage.gr…
mkoeppe Feb 20, 2023
4c437cb
sage.rings: More # optional
mkoeppe Jun 4, 2023
97b5398
More # optional
mkoeppe Jun 9, 2023
d1f5704
sage.rings.finite_rings: More # optional
mkoeppe Jun 28, 2023
a1e4409
sage.rings.finite_rings: ./sage -fixdoctests --only-tags
mkoeppe Jun 28, 2023
a44af4d
sage.schemes: Update # needs
mkoeppe Jun 30, 2023
1f7e48a
Update # optional/needs
mkoeppe Jul 1, 2023
173d3c3
sage.rings: Update # optional / # needs
mkoeppe Jul 2, 2023
c0118ba
src/sage/rings/polynomial/polynomial_ring.py: Use '# needs sage.libs.…
mkoeppe Jul 8, 2023
19222fb
./sage -fixdoctests --distribution sagemath-categories --probe sage.r…
mkoeppe Jul 13, 2023
a04dbbc
src/sage/rings/finite_rings/conway_polynomials.py: Use lazy_import fo…
mkoeppe Jul 15, 2023
e78d4ea
sage.rings: Update # needs
mkoeppe Jul 16, 2023
6042ff7
sage.rings.{finite_rings,polynomial}: Modularization fixes for imports
mkoeppe Jul 17, 2023
1bc26c2
sage.rings.finite_rings: Update # needs
mkoeppe Jul 17, 2023
1b54228
src/sage/rings/finite_rings/conway_polynomials.py: Add # needs
mkoeppe Jul 22, 2023
baca60e
sage.rings: Update # needs
mkoeppe Jul 22, 2023
1810daf
sage.rings.finite_rings: Update # needs
mkoeppe Aug 6, 2023
a3ef387
sage.rings.finite_rings: Update # needs
mkoeppe Aug 7, 2023
a18e1ab
src/sage/rings: sage -fixdoctests --only-tags
mkoeppe Aug 8, 2023
e9630a6
pkgs/sagemath-categories/MANIFEST.in.m4: Add sage.rings.finite_rings.…
mkoeppe Jan 27, 2023
8a823ff
src/sage/rings/finite_rings: Use more block tags
mkoeppe Aug 10, 2023
98283ea
src/sage/rings/finite_rings/conway_polynomials.py: Fix # needs
mkoeppe Aug 11, 2023
744ddf8
src/sage/rings/finite_rings/residue_field.pyx: Make a doctest work wh…
mkoeppe Aug 12, 2023
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sage.rings.finite_rings: ./sage -fixdoctests --only-tags
Matthias Koeppe committed Aug 10, 2023
commit a1e4409baab5475c01bf9913327125ed6d5f2c08
101 changes: 51 additions & 50 deletions src/sage/rings/finite_rings/hom_finite_field.pyx
View file
Original file line number Diff line number Diff line change
@@ -1,3 +1,4 @@
# sage.doctest: optional - sage.rings.finite_rings
"""
Finite field morphisms
@@ -13,8 +14,8 @@ EXAMPLES::
Construction of an embedding::
sage: k.<t> = GF(3^7) # optional - sage.rings.finite_rings
sage: K.<T> = GF(3^21) # optional - sage.rings.finite_rings
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K)); f
Ring morphism:
From: Finite Field in t of size 3^7
@@ -44,8 +45,8 @@ map which is the inverse of `f` on the image of `f`::
There is no embedding of `GF(5^6)` into `GF(5^11)`::
sage: k.<t> = GF(5^6) # optional - sage.rings.finite_rings
sage: K.<T> = GF(5^11) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^6)
sage: K.<T> = GF(5^11)
sage: FiniteFieldHomomorphism_generic(Hom(k, K))
Traceback (most recent call last):
...
@@ -54,7 +55,7 @@ There is no embedding of `GF(5^6)` into `GF(5^11)`::
Construction of Frobenius endomorphisms::
sage: k.<t> = GF(7^14) # optional - sage.rings.finite_rings
sage: k.<t> = GF(7^14)
sage: Frob = k.frobenius_endomorphism(); Frob
Frobenius endomorphism t |--> t^7 on Finite Field in t of size 7^14
sage: Frob(t)
@@ -125,8 +126,8 @@ cdef class SectionFiniteFieldHomomorphism_generic(Section):
TESTS::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7) # optional - sage.rings.finite_rings
sage: K.<T> = GF(3^21) # optional - sage.rings.finite_rings
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: g = f.section()
sage: g(f(t^3+t^2+1))
@@ -153,8 +154,8 @@ cdef class SectionFiniteFieldHomomorphism_generic(Section):
EXAMPLES::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7) # optional - sage.rings.finite_rings
sage: K.<T> = GF(3^21) # optional - sage.rings.finite_rings
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: g = f.section()
sage: g._repr_()
@@ -170,8 +171,8 @@ cdef class SectionFiniteFieldHomomorphism_generic(Section):
EXAMPLES::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7) # optional - sage.rings.finite_rings
sage: K.<T> = GF(3^21) # optional - sage.rings.finite_rings
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: g = f.section()
sage: g._latex_()
@@ -188,8 +189,8 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
TESTS::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7) # optional - sage.rings.finite_rings
sage: K.<T> = GF(3^21) # optional - sage.rings.finite_rings
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: TestSuite(f).run()
@@ -199,16 +200,16 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
TESTS::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7) # optional - sage.rings.finite_rings
sage: K.<T> = GF(3^21) # optional - sage.rings.finite_rings
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K)); f
Ring morphism:
From: Finite Field in t of size 3^7
To: Finite Field in T of size 3^21
Defn: t |--> T^20 + 2*T^18 + T^16 + 2*T^13 + T^9 + 2*T^8 + T^7 + T^6 + T^5 + T^3 + 2*T^2 + T
sage: k.<t> = GF(3^6) # optional - sage.rings.finite_rings
sage: K.<t> = GF(3^9) # optional - sage.rings.finite_rings
sage: k.<t> = GF(3^6)
sage: K.<t> = GF(3^9)
sage: FiniteFieldHomomorphism_generic(Hom(k, K))
Traceback (most recent call last):
...
@@ -249,8 +250,8 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
TESTS::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7) # optional - sage.rings.finite_rings
sage: K.<T> = GF(3^21) # optional - sage.rings.finite_rings
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: g = copy(f)
sage: g.section()(g(t)) == f.section()(f(t))
@@ -282,8 +283,8 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
EXAMPLES::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7) # optional - sage.rings.finite_rings
sage: K.<T> = GF(3^21) # optional - sage.rings.finite_rings
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: f._latex_()
'\\Bold{F}_{3^{7}} \\hookrightarrow \\Bold{F}_{3^{21}}'
@@ -295,8 +296,8 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
TESTS::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^3) # optional - sage.rings.finite_rings
sage: K.<T> = GF(3^9) # optional - sage.rings.finite_rings
sage: k.<t> = GF(3^3)
sage: K.<T> = GF(3^9)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: f(t)
2*T^6 + 2*T^4 + T^2 + T
@@ -323,8 +324,8 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
EXAMPLES::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^3) # optional - sage.rings.finite_rings
sage: K.<T> = GF(3^9) # optional - sage.rings.finite_rings
sage: k.<t> = GF(3^3)
sage: K.<T> = GF(3^9)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: f.is_injective()
True
@@ -340,8 +341,8 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
EXAMPLES::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^3) # optional - sage.rings.finite_rings
sage: K.<T> = GF(3^9) # optional - sage.rings.finite_rings
sage: k.<t> = GF(3^3)
sage: K.<T> = GF(3^9)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: f.is_surjective()
False
@@ -364,8 +365,8 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
EXAMPLES::
sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
sage: k.<t> = GF(3^7) # optional - sage.rings.finite_rings
sage: K.<T> = GF(3^21) # optional - sage.rings.finite_rings
sage: k.<t> = GF(3^7)
sage: K.<T> = GF(3^21)
sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
sage: g = f.section(); g
Section of Ring morphism:
@@ -395,7 +396,7 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
EXAMPLES::
sage: k.<t> = GF(3^7) # optional - sage.rings.finite_rings
sage: k.<t> = GF(3^7)
sage: K.<T>, f = k.extension(3, map=True)
sage: b = f(t^2); b
2*T^20 + 2*T^19 + T^18 + T^15 + 2*T^14 + 2*T^13 + 2*T^12 + T^8 + 2*T^6 + T^5 + 2*T^4 + T^3 + 2*T^2 + T
@@ -417,7 +418,7 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
TESTS::
sage: k.<t> = GF(5^3) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: embed = Frob.fixed_field()[1]
sage: hash(embed) # random
@@ -431,7 +432,7 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
TESTS::
sage: k.<t> = GF(5^3) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: embed = Frob.fixed_field()[1]
sage: embed.__reduce__() # indirect doctest
@@ -459,7 +460,7 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
TESTS::
sage: k.<t> = GF(5^3) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: embed = Frob.fixed_field()[1]
sage: f = loads(dumps(embed))
@@ -481,7 +482,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
TESTS::
sage: k.<a> = GF(7^11) # optional - sage.rings.finite_rings
sage: k.<a> = GF(7^11)
sage: Frob = k.frobenius_endomorphism(5)
sage: TestSuite(Frob).run()
@@ -506,7 +507,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
TESTS::
sage: from sage.rings.finite_rings.hom_finite_field import FrobeniusEndomorphism_finite_field
sage: k.<t> = GF(5^3) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: FrobeniusEndomorphism_finite_field(k)
Frobenius endomorphism t |--> t^5 on Finite Field in t of size 5^3
sage: FrobeniusEndomorphism_finite_field(k, 2)
@@ -543,7 +544,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
EXAMPLES::
sage: k.<t> = GF(5^3) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism(); Frob
Frobenius endomorphism t |--> t^5 on Finite Field in t of size 5^3
@@ -567,7 +568,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
EXAMPLES::
sage: k.<t> = GF(5^3) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism(); Frob
Frobenius endomorphism t |--> t^5 on Finite Field in t of size 5^3
@@ -590,7 +591,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
EXAMPLES::
sage: k.<t> = GF(5^3) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: Frob._latex_()
't \\mapsto t^{5}'
@@ -612,7 +613,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
"""
TESTS::
sage: k.<t> = GF(5^3) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: Frob(t)
2*t^2 + 4*t + 4
@@ -631,7 +632,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
EXAMPLES::
sage: k.<t> = GF(5^12) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^12)
sage: Frob = k.frobenius_endomorphism()
sage: Frob.order()
12
@@ -654,7 +655,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
EXAMPLES::
sage: k.<t> = GF(5^12) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^12)
sage: Frob = k.frobenius_endomorphism()
sage: Frob.power()
1
@@ -672,7 +673,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
EXAMPLES::
sage: k.<t> = GF(5^12) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^12)
sage: Frob = k.frobenius_endomorphism(); Frob
Frobenius endomorphism t |--> t^5 on Finite Field in t of size 5^12
sage: Frob^2
@@ -694,7 +695,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
EXAMPLES::
sage: k.<a> = GF(7^11) # optional - sage.rings.finite_rings
sage: k.<a> = GF(7^11)
sage: f = k.frobenius_endomorphism(5)
sage: (f.inverse() * f).is_identity()
True
@@ -707,7 +708,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
EXAMPLES::
sage: k.<t> = GF(5^12) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^12)
sage: f = k.frobenius_endomorphism(); f
Frobenius endomorphism t |--> t^5 on Finite Field in t of size 5^12
sage: g = k.frobenius_endomorphism(2); g
@@ -745,7 +746,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
EXAMPLES::
sage: k.<t> = GF(5^6) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^6)
sage: f = k.frobenius_endomorphism(2)
sage: kfixed, embed = f.fixed_field()
sage: kfixed
@@ -778,7 +779,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
EXAMPLES::
sage: k.<t> = GF(5^3) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: Frob.is_injective()
True
@@ -793,7 +794,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
EXAMPLES::
sage: k.<t> = GF(5^3) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: Frob.is_surjective()
True
@@ -807,7 +808,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
EXAMPLES::
sage: k.<t> = GF(5^3) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: Frob.is_identity()
False
@@ -822,7 +823,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
EXAMPLES::
sage: k.<t> = GF(5^3) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: hash(Frob) # random
383183030479672104
@@ -835,7 +836,7 @@ cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
TESTS::
sage: k.<t> = GF(5^3) # optional - sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism(2)
sage: Frob
Frobenius endomorphism t |--> t^(5^2) on Finite Field in t of size 5^3
6 changes: 3 additions & 3 deletions src/sage/rings/finite_rings/hom_prime_finite_field.pyx
View file
Original file line number Diff line number Diff line change
@@ -54,15 +54,15 @@ cdef class FiniteFieldHomomorphism_prime(FiniteFieldHomomorphism_generic):
sage: from sage.rings.finite_rings.hom_prime_finite_field import FiniteFieldHomomorphism_prime
sage: k = GF(3)
sage: K.<T> = GF(3^4) # optional - sage.rings.finite_rings
sage: f = FiniteFieldHomomorphism_prime(Hom(k, K)); f
sage: f = FiniteFieldHomomorphism_prime(Hom(k, K)); f # optional - sage.rings.finite_rings
Ring morphism:
From: Finite Field of size 3
To: Finite Field in T of size 3^4
Defn: 1 |--> 1
sage: k.<t> = GF(3^2) # optional - sage.rings.finite_rings
sage: K.<T> = GF(3^4) # optional - sage.rings.finite_rings
sage: f = FiniteFieldHomomorphism_prime(Hom(k, K)); f
sage: f = FiniteFieldHomomorphism_prime(Hom(k, K)); f # optional - sage.rings.finite_rings
Traceback (most recent call last):
...
TypeError: The domain is not a finite prime field
@@ -82,7 +82,7 @@ cdef class FiniteFieldHomomorphism_prime(FiniteFieldHomomorphism_generic):
sage: from sage.rings.finite_rings.hom_prime_finite_field import FiniteFieldHomomorphism_prime
sage: k = GF(3)
sage: K.<t> = GF(3^5) # optional - sage.rings.finite_rings
sage: f = FiniteFieldHomomorphism_prime(Hom(k, K))
sage: f = FiniteFieldHomomorphism_prime(Hom(k, K)) # optional - sage.rings.finite_rings
sage: a = f(4); a
1
sage: a.parent()
67 changes: 34 additions & 33 deletions src/sage/rings/finite_rings/homset.py
View file
Original file line number Diff line number Diff line change
@@ -1,3 +1,4 @@
# sage.doctest: optional - sage.rings.finite_rings
"""
Homset for finite fields
@@ -6,8 +7,8 @@
EXAMPLES::
sage: R.<t> = ZZ[]
sage: E.<a> = GF(25, modulus = t^2 - 2) # optional - sage.rings.finite_rings
sage: F.<b> = GF(625) # optional - sage.rings.finite_rings
sage: E.<a> = GF(25, modulus = t^2 - 2)
sage: F.<b> = GF(625)
sage: H = Hom(E, F)
sage: f = H([4*b^3 + 4*b^2 + 4*b]); f
Ring morphism:
@@ -27,10 +28,10 @@
sage: End(E)
Automorphism group of Finite Field in a of size 5^2
sage: End(GF(7))[0] # optional - sage.rings.finite_rings
sage: End(GF(7))[0]
Ring endomorphism of Finite Field of size 7
Defn: 1 |--> 1
sage: H = Hom(GF(7), GF(49, 'c')) # optional - sage.rings.finite_rings
sage: H = Hom(GF(7), GF(49, 'c'))
sage: H[0](2)
2
"""
@@ -59,8 +60,8 @@ def __call__(self, im_gens, base_map=None, check=True):
EXAMPLES::
sage: R.<t> = ZZ[]
sage: E.<a> = GF(25, modulus = t^2 - 2) # optional - sage.rings.finite_rings
sage: F.<b> = GF(625) # optional - sage.rings.finite_rings
sage: E.<a> = GF(25, modulus = t^2 - 2)
sage: F.<b> = GF(625)
sage: End(E)
Automorphism group of Finite Field in a of size 5^2
sage: list(Hom(E, F))
@@ -74,17 +75,17 @@ def __call__(self, im_gens, base_map=None, check=True):
Defn: a |--> b^3 + b^2 + b]
sage: [phi(2*a)^2 for phi in Hom(E, F)]
[3, 3]
sage: End(GF(7))[0] # optional - sage.rings.finite_rings
sage: End(GF(7))[0]
Ring endomorphism of Finite Field of size 7
Defn: 1 |--> 1
sage: H = Hom(GF(7), GF(49, 'c')) # optional - sage.rings.finite_rings
sage: H = Hom(GF(7), GF(49, 'c'))
sage: H[0](2)
2
sage: Hom(GF(49, 'c'), GF(7)).list() # optional - sage.rings.finite_rings
sage: Hom(GF(49, 'c'), GF(7)).list()
[]
sage: Hom(GF(49, 'c'), GF(81, 'd')).list() # optional - sage.rings.finite_rings
sage: Hom(GF(49, 'c'), GF(81, 'd')).list()
[]
sage: H = Hom(GF(9, 'a'), GF(81, 'b')) # optional - sage.rings.finite_rings
sage: H = Hom(GF(9, 'a'), GF(81, 'b'))
sage: H == loads(dumps(H))
True
"""
@@ -116,8 +117,8 @@ def _coerce_impl(self, x):
EXAMPLES::
sage: k.<a> = GF(25) # optional - sage.rings.finite_rings
sage: l.<b> = GF(625) # optional - sage.rings.finite_rings
sage: k.<a> = GF(25)
sage: l.<b> = GF(625)
sage: H = Hom(k, l)
sage: G = loads(dumps(H))
sage: H is G
@@ -142,11 +143,11 @@ def _repr_(self):
EXAMPLES::
sage: Hom(GF(4, 'a'), GF(16, 'b'))._repr_() # optional - sage.rings.finite_rings
sage: Hom(GF(4, 'a'), GF(16, 'b'))._repr_()
'Set of field embeddings from Finite Field in a of size 2^2 to Finite Field in b of size 2^4'
sage: Hom(GF(4, 'a'), GF(4, 'c'))._repr_() # optional - sage.rings.finite_rings
sage: Hom(GF(4, 'a'), GF(4, 'c'))._repr_()
'Set of field embeddings from Finite Field in a of size 2^2 to Finite Field in c of size 2^2'
sage: Hom(GF(4, 'a'), GF(4, 'a'))._repr_() # optional - sage.rings.finite_rings
sage: Hom(GF(4, 'a'), GF(4, 'a'))._repr_()
'Automorphism group of Finite Field in a of size 2^2'
"""
D = self.domain()
@@ -162,11 +163,11 @@ def is_aut(self):
EXAMPLES::
sage: Hom(GF(4, 'a'), GF(16, 'b')).is_aut() # optional - sage.rings.finite_rings
sage: Hom(GF(4, 'a'), GF(16, 'b')).is_aut()
False
sage: Hom(GF(4, 'a'), GF(4, 'c')).is_aut() # optional - sage.rings.finite_rings
sage: Hom(GF(4, 'a'), GF(4, 'c')).is_aut()
False
sage: Hom(GF(4, 'a'), GF(4, 'a')).is_aut() # optional - sage.rings.finite_rings
sage: Hom(GF(4, 'a'), GF(4, 'a')).is_aut()
True
"""
return self.domain() == self.codomain()
@@ -177,12 +178,12 @@ def order(self):
EXAMPLES::
sage: K.<a> = GF(125) # optional - sage.rings.finite_rings
sage: K.<a> = GF(125)
sage: End(K)
Automorphism group of Finite Field in a of size 5^3
sage: End(K).order()
3
sage: L.<b> = GF(25) # optional - sage.rings.finite_rings
sage: L.<b> = GF(25)
sage: Hom(L, K).order() == Hom(K, L).order() == 0
True
"""
@@ -200,7 +201,7 @@ def __len__(self):
EXAMPLES::
sage: K.<a> = GF(25) # optional - sage.rings.finite_rings
sage: K.<a> = GF(25)
sage: len(End(K))
2
"""
@@ -212,15 +213,15 @@ def list(self):
EXAMPLES::
sage: K.<a> = GF(25) # optional - sage.rings.finite_rings
sage: K.<a> = GF(25)
sage: End(K)
Automorphism group of Finite Field in a of size 5^2
sage: list(End(K))
[Ring endomorphism of Finite Field in a of size 5^2
Defn: a |--> 4*a + 1,
Ring endomorphism of Finite Field in a of size 5^2
Defn: a |--> a]
sage: L.<z> = GF(7^6) # optional - sage.rings.finite_rings
sage: L.<z> = GF(7^6)
sage: [g for g in End(L) if (g^3)(z) == z]
[Ring endomorphism of Finite Field in z of size 7^6
Defn: z |--> z,
@@ -231,8 +232,8 @@ def list(self):
Between isomorphic fields with different moduli::
sage: k1 = GF(1009) # optional - sage.rings.finite_rings
sage: k2 = GF(1009, modulus="primitive") # optional - sage.rings.finite_rings
sage: k1 = GF(1009)
sage: k2 = GF(1009, modulus="primitive")
sage: Hom(k1, k2).list()
[
Ring morphism:
@@ -248,8 +249,8 @@ def list(self):
Defn: 11 |--> 11
]
sage: k1.<a> = GF(1009^2, modulus="first_lexicographic") # optional - sage.rings.finite_rings
sage: k2.<b> = GF(1009^2, modulus="conway") # optional - sage.rings.finite_rings
sage: k1.<a> = GF(1009^2, modulus="first_lexicographic")
sage: k2.<b> = GF(1009^2, modulus="conway")
sage: Hom(k1, k2).list()
[
Ring morphism:
@@ -266,7 +267,7 @@ def list(self):
Check that :trac:`11390` is fixed::
sage: K = GF(1<<16,'a'); L = GF(1<<32,'b') # optional - sage.rings.finite_rings
sage: K = GF(1<<16,'a'); L = GF(1<<32,'b')
sage: K.Hom(L)[0]
Ring morphism:
From: Finite Field in a of size 2^16
@@ -294,7 +295,7 @@ def __getitem__(self, n):
"""
EXAMPLES::
sage: H = Hom(GF(32, 'a'), GF(1024, 'b')) # optional - sage.rings.finite_rings
sage: H = Hom(GF(32, 'a'), GF(1024, 'b'))
sage: H[1]
Ring morphism:
From: Finite Field in a of size 2^5
@@ -320,7 +321,7 @@ def index(self, item):
EXAMPLES::
sage: K.<z> = GF(1024) # optional - sage.rings.finite_rings
sage: K.<z> = GF(1024)
sage: g = End(K)[3]
sage: End(K).index(g) == 3
True
@@ -333,13 +334,13 @@ def _an_element_(self):
TESTS::
sage: Hom(GF(3^3, 'a'), GF(3^6, 'b')).an_element() # optional - sage.rings.finite_rings
sage: Hom(GF(3^3, 'a'), GF(3^6, 'b')).an_element()
Ring morphism:
From: Finite Field in a of size 3^3
To: Finite Field in b of size 3^6
Defn: a |--> 2*b^5 + 2*b^4
sage: Hom(GF(3^3, 'a'), GF(3^2, 'c')).an_element() # optional - sage.rings.finite_rings
sage: Hom(GF(3^3, 'a'), GF(3^2, 'c')).an_element()
Traceback (most recent call last):
...
EmptySetError: no homomorphisms from Finite Field in a of size 3^3 to Finite Field in c of size 3^2