galois.FieldArray.minimal_poly

galois.FieldArray.minimal_poly() → Poly

Computes the minimal polynomial of a finite field element $a$ .

Returns:¶

For scalar inputs, the minimal polynomial $m_{a} (x)$ of $a$ over $GF (p)$ .

Raises:¶

NotImplementedError – If the array is a a square $n \times n$ matrix (2-D array).
ValueError – If the array is not a single finite field element (scalar 0-D array).

Notes¶

An element $a$ of $GF (p^{m})$ has minimal polynomial $m_{a} (x)$ over $GF (p)$ . The minimal polynomial when evaluated in $GF (p^{m})$ annihilates $a$ , that is $m_{a} (a) = 0$ . The minimal polynomial always divides the characteristic polynomial. In prime fields $GF (p)$ , the minimal polynomial of $a$ is simply $m_{a} (x) = x - a$ .

References¶

Examples¶

The minimal polynomial of the element $a$ .

Integer Polynomial Power

In [1]: GF = galois.GF(3**5)

In [2]: a = GF.Random(); a
Out[2]: GF(68, order=3^5)

In [3]: poly = a.minimal_poly(); poly
Out[3]: Poly(x^5 + 2x^4 + x^3 + 2x^2 + 2x + 2, GF(3))

# The minimal polynomial annihilates a
In [4]: poly(a, field=GF)
Out[4]: GF(0, order=3^5)

# The minimal polynomial always divides the characteristic polynomial
In [5]: divmod(a.characteristic_poly(), poly)
Out[5]: (Poly(1, GF(3)), Poly(0, GF(3)))

In [6]: GF = galois.GF(3**5, repr="poly")

In [7]: a = GF.Random(); a
Out[7]: GF(α^4 + 2α^2 + 2α, order=3^5)

In [8]: poly = a.minimal_poly(); poly
Out[8]: Poly(x^5 + 2x^4 + 2x^3 + 2, GF(3))

# The minimal polynomial annihilates a
In [9]: poly(a, field=GF)
Out[9]: GF(0, order=3^5)

# The minimal polynomial always divides the characteristic polynomial
In [10]: divmod(a.characteristic_poly(), poly)
Out[10]: (Poly(1, GF(3)), Poly(0, GF(3)))

In [11]: GF = galois.GF(3**5, repr="power")

In [12]: a = GF.Random(); a
Out[12]: GF(α^40, order=3^5)

In [13]: poly = a.minimal_poly(); poly
Out[13]: Poly(x^5 + x^4 + 2, GF(3))

# The minimal polynomial annihilates a
In [14]: poly(a, field=GF)
Out[14]: GF(0, order=3^5)

# The minimal polynomial always divides the characteristic polynomial
In [15]: divmod(a.characteristic_poly(), poly)
Out[15]: (Poly(1, GF(3)), Poly(0, GF(3)))