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Let \(F\) be a subfield of a field \(E\). Let \(f(x)\in F[x]\). Suppose that \(\alpha\in E\) is an element such that \(f(\alpha)=0\). Then \(\alpha\) is a zero of \(f(x)\) in \(E\).
Example
Let \(F=\mathbb{Q}, E=\mathbb{R}\) and \(f(x)=x^2-2\). Then \(\pm\sqrt{2}\) are zeros of \(f(x)\) in \(\mathbb{R}\).
An element \(a\in \mathbb{F}\) is a zero of \(f(x) \in \mathbb{F}[x]\) iff \(x-a\) is a factor of \(f(x)\) in \(\mathbb{F}[x]\).
Proof
Use the division algorithm.
A non-zero polynomial \(f(x)\in \mathbb{F}[x]\) of degree \(n\) can have at most \(n\) distinct zeros in \(\mathbb{F}\).
Proof
Let \(\mathbb{F}\) be a field of order \(p\). If \(G\) is a finite subgroup of the multiplicative group \(\mathbb{F}^\times\), then \(G\) is cyclic. In particular, if \(\mathbb{F}\) is a finite field, then \(\mathbb{F}^{\times}\) is cyclic.
\(G\) 當然是 abelian。
Proof
By the fundamental theorem of finitely generated abelian groups, the finite abelian group \(G\) is isomorphic to \(\mathbb{Z}_{p_1^{e_1}}\times\cdots\times\mathbb{Z}_{p_k^{e_k}}\) for some primes \(p_i\) and integers \(e_i\ge 1\).
Note that \(G\) contains a subgroup \(H\) isomorphic to \(\mathbb{Z}_{p_1}\times\cdots\times\mathbb{Z}_{p_k}\). Suppose \(p_i=p_j=p\), then \(x^p = 1\) has at least \(p^2\) solutions in \(H\subset \mathbb{F}\), which, by the above corollary, is a contradiction.
Therefore, all \(p_i\) are distinct and by this theorem, \(G\) is cyclic.
\(p^2\) 個解哪裡來？例如 \(x=(a,b,1,\cdots,1)\)，而 \(a\) 和 \(b\) 各有 \(p\) 種選擇使 \(x^p = 1\)。
A nonconstant polynomial \(f(x) \in \mathbb{F}[x]\) is irreducible over \(\mathbb{F}\) or is an irreducible polynomial in \(\mathbb{F}[x]\) if \(f(x)\) cannot be expressed as a product \(g(x)h(x)\) of two polynoials in \(\mathbb{F}[x]\) with \(0< \deg{g(x)},\deg{h(x)} < \deg{f(x)}\) simultaneously.
If a polynomial \(f(x)\in\mathbb{F}[x]\) is nonconstant and is not irreducible over \(\mathbb{F}\), then it is reducible over \(\mathbb{F}\).
Examples
是否 reducible 端視 over 哪個 field！
The definition of irreducible polynomials can also be given as
\(f(x)\) is irreducible over \(\mathbb{F}\) if for any factorization \(f(x)=g(x)h(x)\) in \(\mathbb{F}[x]\), one of \(g(x)\) and \(h(x)\) is a unit.
因為 \(g\) 或 \(h\) 其中必有一 constant polynomial，而 \(\mathbb{F}[x]\) 中只有 constants 是 unit。
Suppose that \(f(x)\in\mathbb{F}[x]\) is of degree \(2\) or \(3\). Then \(f(x)\) is rducible over \(\mathbb{F}\) iff \(f(x)\) has a zero in \(\mathbb{F}\).