# Base class for finite field elements¶

Base class for finite field elements

AUTHORS:

- David Roe (2010-1-14) -- factored out of sage.structure.element

class sage.rings.finite_rings.element_base.FinitePolyExtElement

Elements represented as polynomials modulo a given ideal.

TESTS:

sage: k.<a> = GF(64)
sage: TestSuite(a).run()


Return the additive order of this finite field element.

EXAMPLES:

sage: k.<a> = FiniteField(2^12, 'a')
sage: b = a^3 + a + 1
2
1

charpoly(var='x', algorithm='matrix')

Return the characteristic polynomial of self as a polynomial with given variable.

INPUT:

• var - string (default: ‘x’)
• algorithm - string (default: ‘matrix’)
• ‘matrix’ - return the charpoly computed from the matrix of left multiplication by self
• ‘pari’ – use pari’s charpoly routine on polymods, which is not very good except in small cases

The result is not cached.

EXAMPLES:

sage: k.<a> = GF(19^2)
sage: parent(a)
Finite Field in a of size 19^2
sage: a.charpoly('X')
X^2 + 18*X + 2
sage: a^2 + 18*a + 2
0
sage: a.charpoly('X', algorithm='pari')
X^2 + 18*X + 2

frobenius(k=1)

Return the $$(p^k)^{th}$$ power of self, where $$p$$ is the characteristic of the field.

INPUT:

• k - integer (default: 1, must fit in C int type)

Note that if $$k$$ is negative, then this computes the appropriate root.

EXAMPLES:

sage: F.<a> = GF(29^2)
sage: z = a^2 + 5*a + 1
sage: z.pth_power()
19*a + 20
sage: z.pth_power(10)
10*a + 28
sage: z.pth_power(-10) == z
True
sage: F.<b> = GF(2^12)
sage: y = b^3 + b + 1
sage: y == (y.pth_power(-3))^(2^3)
True
sage: y.pth_power(2)
b^7 + b^6 + b^5 + b^4 + b^3 + b

is_square()

Returns True if and only if this element is a perfect square.

EXAMPLES:

sage: k.<a> = FiniteField(9, impl='givaro', modulus='primitive')
sage: a.is_square()
False
sage: (a**2).is_square()
True
sage: k.<a> = FiniteField(4, impl='ntl', modulus='primitive')
sage: (a**2).is_square()
True
sage: k.<a> = FiniteField(17^5, impl='pari_ffelt', modulus='primitive')
sage: a.is_square()
False
sage: (a**2).is_square()
True

sage: k(0).is_square()
True

minimal_polynomial(var='x')

Returns the minimal polynomial of this element (over the corresponding prime subfield).

EXAMPLES:

sage: k.<a> = FiniteField(3^4)
sage: parent(a)
Finite Field in a of size 3^4
sage: b=a**20;p=charpoly(b,"y");p
y^4 + 2*y^2 + 1
sage: factor(p)
(y^2 + 1)^2
sage: b.minimal_polynomial('y')
y^2 + 1

minpoly(var='x')

Returns the minimal polynomial of this element (over the corresponding prime subfield).

EXAMPLES:

sage: k.<a> = FiniteField(19^2)
sage: parent(a)
Finite Field in a of size 19^2
sage: b=a**20;p=b.charpoly("x");p
x^2 + 15*x + 4
sage: factor(p)
(x + 17)^2
sage: b.minpoly('x')
x + 17

multiplicative_order()

Return the multiplicative order of this field element.

EXAMPLE:

sage: S.<a> = GF(5^3); S
Finite Field in a of size 5^3
sage: a.multiplicative_order()
124
sage: (a^8).multiplicative_order()
31
sage: S(0).multiplicative_order()
Traceback (most recent call last):
...
ArithmeticError: Multiplicative order of 0 not defined.

norm()

Return the norm of self down to the prime subfield.

This is the product of the Galois conjugates of self.

EXAMPLES:

sage: S.<b> = GF(5^2); S
Finite Field in b of size 5^2
sage: b.norm()
2
sage: b.charpoly('t')
t^2 + 4*t + 2


Next we consider a cubic extension:

sage: S.<a> = GF(5^3); S
Finite Field in a of size 5^3
sage: a.norm()
2
sage: a.charpoly('t')
t^3 + 3*t + 3
sage: a * a^5 * (a^25)
2

nth_root(n, extend=False, all=False, algorithm=None, cunningham=False)

Returns an $$n$$th root of self.

INPUT:

• n - integer $$\geq 1$$
• extend - bool (default: False); if True, return an $$n$$th root in an extension ring, if necessary. Otherwise, raise a ValueError if the root is not in the base ring. Warning: this option is not implemented!
• all - bool (default: False); if True, return all $$n$$th roots of self, instead of just one.
• algorithm - string (default: None); ‘Johnston’ is the only currently supported option. For IntegerMod elements, the problem is reduced to the prime modulus case using CRT and $$p$$-adic logs, and then this algorithm used.

OUTPUT:

If self has an $$n$$th root, returns one (if all is False) or a list of all of them (if all is True). Otherwise, raises a ValueError (if extend is False) or a NotImplementedError (if extend is True).

Warning

The extend option is not implemented (yet).

EXAMPLES:

sage: K = GF(31)
sage: a = K(22)
sage: K(22).nth_root(7)
13
sage: K(25).nth_root(5)
5
sage: K(23).nth_root(3)
29

sage: K.<a> = GF(625)
sage: (3*a^2+a+1).nth_root(13)**13
3*a^2 + a + 1

sage: k.<a> = GF(29^2)
sage: b = a^2 + 5*a + 1
sage: b.nth_root(11)
3*a + 20
sage: b.nth_root(5)
Traceback (most recent call last):
...
ValueError: no nth root
sage: b.nth_root(5, all = True)
[]
sage: b.nth_root(3, all = True)
[14*a + 18, 10*a + 13, 5*a + 27]

sage: k.<a> = GF(29^5)
sage: b = a^2 + 5*a + 1
sage: b.nth_root(5)
19*a^4 + 2*a^3 + 2*a^2 + 16*a + 3
sage: b.nth_root(7)
Traceback (most recent call last):
...
ValueError: no nth root
sage: b.nth_root(4, all=True)
[]


TESTS:

sage: for p in [2,3,5,7,11]:  # long time, random because of PARI warnings
....:     for n in [2,5,10]:
....:         q = p^n
....:         K.<a> = GF(q)
....:         for r in (q-1).divisors():
....:             if r == 1: continue
....:             x = K.random_element()
....:             y = x^r
....:             assert y.nth_root(r)^r == y
....:             assert (y^41).nth_root(41*r)^(41*r) == y^41
....:             assert (y^307).nth_root(307*r)^(307*r) == y^307
sage: k.<a> = GF(4)
sage: a.nth_root(0,all=True)
[]
sage: k(1).nth_root(0,all=True)
[a, a + 1, 1]


ALGORITHMS:

• The default is currently an algorithm described in the following paper:

Johnston, Anna M. A generalized qth root algorithm. Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms. Baltimore, 1999: pp 929-930.

AUTHOR:

• David Roe (2010-02-13)
pth_power(k=1)

Return the $$(p^k)^{th}$$ power of self, where $$p$$ is the characteristic of the field.

INPUT:

• k - integer (default: 1, must fit in C int type)

Note that if $$k$$ is negative, then this computes the appropriate root.

EXAMPLES:

sage: F.<a> = GF(29^2)
sage: z = a^2 + 5*a + 1
sage: z.pth_power()
19*a + 20
sage: z.pth_power(10)
10*a + 28
sage: z.pth_power(-10) == z
True
sage: F.<b> = GF(2^12)
sage: y = b^3 + b + 1
sage: y == (y.pth_power(-3))^(2^3)
True
sage: y.pth_power(2)
b^7 + b^6 + b^5 + b^4 + b^3 + b

pth_root(k=1)

Return the $$(p^k)^{th}$$ root of self, where $$p$$ is the characteristic of the field.

INPUT:

• k - integer (default: 1, must fit in C int type)

Note that if $$k$$ is negative, then this computes the appropriate power.

EXAMPLES:

sage: F.<b> = GF(2^12)
sage: y = b^3 + b + 1
sage: y == (y.pth_root(3))^(2^3)
True
sage: y.pth_root(2)
b^11 + b^10 + b^9 + b^7 + b^5 + b^4 + b^2 + b

sqrt(extend=False, all=False)

See :meth:square_root().

EXAMPLES:

sage: k.<a> = GF(3^17)
sage: (a^3 - a - 1).sqrt()
a^16 + 2*a^15 + a^13 + 2*a^12 + a^10 + 2*a^9 + 2*a^8 + a^7 + a^6 + 2*a^5 + a^4 + 2*a^2 + 2*a + 2

square_root(extend=False, all=False)

The square root function.

INPUT:

• extend – bool (default: True); if True, return a square root in an extension ring, if necessary. Otherwise, raise a ValueError if the root is not in the base ring.

Warning

This option is not implemented!

• all - bool (default: False); if True, return all square roots of self, instead of just one.

Warning

The 'extend' option is not implemented (yet).

EXAMPLES:

sage: F = FiniteField(7^2, 'a')
sage: F(2).square_root()
4
sage: F(3).square_root()
2*a + 6
sage: F(3).square_root()**2
3
sage: F(4).square_root()
2
sage: K = FiniteField(7^3, 'alpha', impl='pari_ffelt')
sage: K(3).square_root()
Traceback (most recent call last):
...
ValueError: must be a perfect square.

trace()

Return the trace of this element, which is the sum of the Galois conjugates.

EXAMPLES:

sage: S.<a> = GF(5^3); S
Finite Field in a of size 5^3
sage: a.trace()
0
sage: a.charpoly('t')
t^3 + 3*t + 3
sage: a + a^5 + a^25
0
sage: z = a^2 + a + 1
sage: z.trace()
2
sage: z.charpoly('t')
t^3 + 3*t^2 + 2*t + 2
sage: z + z^5 + z^25
2

class sage.rings.finite_rings.element_base.FiniteRingElement

INPUT:

• parent - a SageObject
sage.rings.finite_rings.element_base.is_FiniteFieldElement(x)

Returns if x is a finite field element.

EXAMPLE:

sage: from sage.rings.finite_rings.element_base import is_FiniteFieldElement
sage: is_FiniteFieldElement(1)
False
sage: is_FiniteFieldElement(IntegerRing())
False
sage: is_FiniteFieldElement(GF(5)(2))
True


Finite Fields

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