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ANALOGIES WITH POLYNOMIALS
KEITH CONRAD
- The Basic Analogies
Similarities between Z and F [T ] are an important theme in number theory. The following table collects some analogous concepts in Z and in F [T ].
Z F [T ] Similarity ± 1 nonzero constants these are the units prime irreducible have only trivial factors |n| deg f role in division theorem positive monic (lead. coeff = 1) standard unit multiple A polynomial is called monic when it has leading coefficient 1, such as T 2 + 7T + 3 but not 2T 2 + 5T − 1. Every nonzero integer has exactly one positive unit multiple, while every nonzero polynomial in F [T ] has exactly one monic unit multiple: just multiply through the polynomial by the inverse of the leading coefficient.
Example 1.1. In Q[T ], the monic unit multiple of 2T 2 + 5T − 1 is 12 (2T 2 + 5T − 1) = T 2 + 52 T − 12. In F 7 , 2 · 4 = 1, so the monic unit multiple of 2T 2 + 5T − 1 in F 7 [T ] is 4(2T 2 + 5T − 1) = T 2 + 6T + 3.
Positive integers are closed under multiplication and monic polynomials are closed under multiplication. Positive integers are also closed under addition but monic polynomials are not generally closed under addition. This is an important difference! The standard notation for a prime number in Z is p. The standard notation for an irreducible polynomial in F [T ] is π = π(T ). (This has nothing to do with 3. 14159... , of course.) By definition, a prime in Z is a number which is not ±1 and its only factors are ±1 and ± itself. Similarly, a polynomial in F [T ] is called irreducible when it is nonconstant (that is, is not a unit) and its only factors are nonzero constants and nonzero constant multiples of itself. Here are some analogous results in Z and F [T ]: (1) In Z, |mn| = |m||n|. In F [T ], deg f g = deg f + deg g. (2) The units in Z have absolute value 1 (which is the smallest absolute value possible for nonzero integers) and the units in F [T ] have degree 0 (the smallest degree possible for nonzero polynomials). (3) In Z if a|b then |a| ≤ |b|. In F [T ], if f |g then deg f ≤ deg g. (4) If a|b and b|a in Z then a = ±b, while if f |g and g|f in F [T ] then f = cg for some nonzero constant c. (5) Every integer other than 0 and ±1 is a product of primes (allowing negative primes!), while every polynomial in F [T ] other than a constant is a product of irreducible polynomials. The most important similarity between Z and F [T ] is the division theorem in both settings. We state them without proof, using similar wording. 1
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Theorem 1.2. For a, b ∈ Z with b 6 = 0, there are unique q and r in Z such that a = bq + r with 0 ≤ r < |b|.
Theorem 1.3. For f, g ∈ F [T ] with g 6 = 0, there are unique q and r in F [T ] such that f = gq + r with r = 0 or deg r < deg g.
The greatest common divisor of two integers is the common divisor largest in size (so always positive). In F [T ], the greatest common divisor of two polynomials is the common monic polynomial factor with the largest degree. Examples will be worked out in the next section. Two integers are called relatively prime when their only common factors are ±1. Simi- larly, two polynomials in F [T ] are called relatively prime when their only common factors are nonzero constants. In both Z and F [T ], relative primality means the only common factors are units. Euclid’s algorithm is the standard method to compute greatest common divisors in Z (so, in particular, to determine relative primality) while a variant of Euclid’s algorithm in F [T ] will perform the same role for polynomials. The standard chain of reasoning div. thm. Euclid Bezout if p|ab then p|a or p|b unique factn
in Z carries over to F [T ] nearly verbatim, with only minor changes needed in most proofs:
div. thm. Euclid Bezout if π|f g then π|f or π|g unique factn. There is one important difference between Z and F [T ]. Division in Z involves remainders ≥ 0, so if two integers are relatively prime Euclid’s algorithm will always have last nonzero remainder 1. But this is false with polynomials: the last nonzero remainder in Euclid’s algorithm for polynomials might be a nonzero constant other than 1, so writing an F [T ]- linear combination of relatively prime polynomials as 1 can involve some additional scaling which we don’t have to do in Z.
Example 1.4. In R[T ], let f (T ) = T 2 + 1 and g(T ) = T − 1. Certainly f (T ) and g(T ) are relatively prime: they have no common factor in R[T ] other than nonzero constants. When we carry out Euclid’s algorithm on these two polynomials we find
T 2 + 1 = (T − 1)(T + 1) + 2
T − 1 = 2
T −
so the last nonzero remainder is 2. This is a nonzero constant in R[T ] but it is not 1. By convention we normalize the gcd of two polynomials to be monic, so the gcd of T 2 + 1 and T − 1 is called 1, not 2.
- Euclid and Bezout: examples Bezout’s identity in Z says for a and b in Z that we can write ax + by = (a, b)
for some integers x and y. Values for x and y can be found by using back-substitution into Euclid’s algorithm for a and b. Similarly, Bezout’s identity for F [T ] says for f (T ) and g(T ) in F [T ] that f (T )u(T ) + g(T )v(T ) = (f, g),
for some u(T ) and v(T ) in F [T ]. Here too the polynomials u(T ) and v(T ) can be found using back-susbtitution into Euclid’s algorithm for f (T ) and g(T ).
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and we stop since we have reached a nonzero constant, 4. The gcd of f and g in F 7 [T ] is 1. The following table summarizes our computations. We list both the last nonzero remain- der in Euclid’s algorithm and the gcd to reinforce the fact that these need not be the same: (f, g) is the monic unit multiple of the last nonzero remainder.
F [T ] Last Remainder (f, g) Q[T ] 1395/256 1 F 2 [T ] 1 1 F 3 [T ] 1 1 F 5 [T ] 4 T + 1 T + 4 F 7 [T ] 4 1 Table 1. f (T ) = T 4 + T 3 + T 2 + T + 1, g(T ) = T 3 − 2 T − 4
Remark 2.2. The case of F 5 [T ] sticks out: here we have a nonconstant gcd, but for the rest the gcd is 1. It turns out there is exactly one other p for which f (T ) and g(T ) are not relatively prime in Fp[T ]: in F 31 [T ], f (T ) and g(T ) have gcd T − 2.
Now let’s realize Bezout’s identity from each of those gcd calculations. We can back- substitute into Euclid’s algorithm to write the last nonzero remainder as a polynomial-linear combination of f and g. In Q[T ], the gcd of f (T ) and g(T ) is found as follows: 1395 256
= (3T 2 + 7T + 5) −
T −
T +
= (3T 2 + 7T + 5) −
(T 3 − 2 T − 4) − (3T 2 + 7T + 5)
T −
T +
= (3T 2 + 7T + 5)
T 2 +
T −
− (T 3 − 2 T − 4)
T +
= f ·
T 2 +
T −
T 3 −
T 2 +
T −
Multiplying through by 256/1395,
1 = f ·
T 2 +
T −
T 3 −
T 2 +
T −
Remark 2.3. The common denominator 155 appearing in this equation factors as 5 · 31. This is related to the special roles of 5 and 31 in Remark 2.2!
In F 2 [T ] we get by back-substitution from Euclid’s algorithm
1 = g − (T 2 + T + 1)(T + 1) = g − (f − g(T + 1))(T + 1)) = f · (T + 1) + g · (1 + (T + 1)(T + 1)) = f · (T + 1) + g · T 2.
ANALOGIES WITH POLYNOMIALS 5
Using back-substitution in F 3 [T ], 1 = g − (T + 2)(T 2 + x + 2) = g − (f − g(T + 1))(T 2 + T + 2) = f · (2T 2 + 2T + 1) + g · (1 + (T + 1)(T 2 + T + 2)) = f · (2T 2 + 2T + 1) + g · (T 3 + 2T 2 ). In F 5 [T ], 4 T + 1 = g − (3T 2 + 2T )(2T + 2) = g − (f − g(T + 1)(2T + 2)) = f · (3T + 3) + g · (1 + (T + 1)(2T + 2)) = f · (3T + 3) + g · (2T 2 + 4T + 3).
The gcd we found in Euclid’s algorithm, 4T + 1, is not monic. To write the monic gcd of f and g as an F 5 [T ]-linear combination of f and g we simply multiply through the equations by −1 = 4: T + 4 = f · (2T + 2) + g · (3T 2 + T + 2). In F 7 [T ], 4 = (3T 2 + 5) − (T + 3)(3T + 5) = (3T 2 + 5) − (g − (3T 2 + 5)(5T ))(3T + 5) = (3T 2 + 5)(1 + 5T (3T + 5)) + g(4T + 2) = (3T 2 + 5)(T 2 + 4T + 1) + g(4T + 2) = (f − g(T + 1))(T 2 + 4T + 1) + g(4T + 2) = f · (T 2 + 4T + 1) + g · (6T 3 + 2T 2 + 6T + 1).
Multiplying through by 4−^1 = 2,
1 = f · (2T 2 + T + 2) + g · (5T 3 + 4T 2 + 5T + 2). We can also do back-substitution in Q[T ]. The calculations would be quite tedious to do by hand on account of the large fractions arising in Euclid’s algorithm. The result will express 1395256 as a Q[T ]-linear combination of f (T ) and g(T ), and then we have to multiply through by the reciprocal 1395256 to write 1 as a Q[T ]-linear combination of f (T ) and g(T ). Omitting the intermediate details, the final result is
1 = f ·
T 2 +
T −
T 3 −
T 2 +
T −
Remark 2.4. The common denominator 155 appearing in this equation factors as 5 · 31. This is related to the special roles of 5 and 31 in Remark 2.2!
- Solving simultaneous congruences: an example In Z, if we want to solve the pair of congruence conditions x ≡ 2 mod 5, x ≡ 11 mod 19,
we lift the first congruence to Z in the form x = 2 + 5y for some y ∈ Z and substitute that into the second congruence and solve for y:
2 + 5y ≡ 11 mod 19 ⇒ 5 y ≡ 9 mod 19 ⇒ y ≡ 17 mod 19.
ANALOGIES WITH POLYNOMIALS 7
Therefore T 4 + 2T 3 + 2T 2 + 1 works, and more generally any polynomial congruent to T 4 + 2T 3 + 2T 2 + 1 mod (T 2 + 1)T 3 works. This is the complete set of solutions to both congruences.
