I'm trying to follow proof of the following theorem in my book.
Theorem 2.2. Let f denote a polynomial with integral coefficients. If a \equiv b \pmod{m} then f(a) \equiv f(b)\pmod {m}.
I understand the proof fine, except for the bit I've coloured below:
Proof: We can suppose f(x) = c_nx^n + c_{n-1}x^{n-1} + ... +c_{0} where the c_i
are integers. Since a \equiv b (\pmod m) we can apply Theorem 2.1, part 4,
repeatedly to find a^2 \equiv b^2, a^3 \equiv b^3, \cdots, a^n \equiv b^n \pmod m, and then
\color{blue}{c_ja^j \equiv c_jb^j\pmod{m}} and finally c_nba^n+c_{n-1}a^{n-1}+\cdots+c_0 \equiv c_nb^n+c_{n-1}b^{n-1}+\cdots+c_0 by Theorem 2.1 part 3.
I don't understand what gives us that {c_ja^j \equiv c_jb^j\pmod{m}}? In fact, I'd expect that {c_ja^j=c_jb^j\pmod{c_jm}} for any c_j > 0.
2.1 part 3 If a \equiv b \pmod{m} and c \equiv d \pmod{m} then a+c \equiv b+d \pmod{m}
2.1 part 4 If a \equiv b \pmod{m} and c \equiv d \pmod{m} then ac \equiv bd \pmod{m}.
Answer
c_j \equiv c_j \pmod m and a^j \equiv b^j \pmod m (from Theorem 2.1 part 4). Using Theorem 2.1 part 4 on these two gives the required result.
No comments:
Post a Comment