2 Discriminants of number fields

We recall basic facts about the discriminant.

Lemma 2.1 ✓

Let $K$ be a number field, $α \in K$ and let $σ_{i}$ be the embeddings of $K$ into $C$ . Then

N_{K / Q} (α) = \prod_{i} σ_{i} (α)

Proof ▼

The proof is standard.

Lemma 2.2 ✓

Let $K$ be a number field, $α \in K$ and let $σ_{i}$ be the embeddings of $K$ into $C$ . Then

{Tr}_{K / Q} (α) = \sum_{i} σ_{i} (α) .

Proof ▼

The proof is standard.

Definition 2.3 ✓

Let $A, K$ be commutative rings with $K$ and $A$ -algebra. let $B = {b_{1}, \dots, b_{n}}$ be a set of elements in $K$ . The discriminant of $B$ is defined as

Δ (B) = det (\begin{matrix} {Tr}_{K / A} (b_{1} b_{1}) & \dots & {Tr}_{K / A} (b_{1} b_{n}) \\ ⋮ & ⋮ \\ {Tr}_{K / A} (b_{n} b_{1}) & \dots & {Tr}_{K / A} (b_{n} b_{n}) \end{matrix}) .

Lemma 2.4 ✓

Let $L / K$ be an extension of fields and let $B = {b_{1}, \dots, b_{n}}$ be a $K$ -basis of $L$ . Then $Δ (B) \neq 0$ .

Proof ▼

The proof is standard.

Lemma 2.5 ✓

Let $K$ be a number field and $B, B^{'}$ bases for $K / Q$ . If $P$ denotes the change of basis matrix, then

Δ (B) = det (P)^{2} Δ (B^{'}) .

Proof ▼

The proof is standard.

Lemma 2.6 ✓

Let $K$ be a number field with basis $B = {b_{1}, \dots, b_{n}}$ and let $σ_{1}, \dots, σ_{n}$ be the embeddings of $K$ into $C$ . Now let $M$ be the matrix

(\begin{matrix} σ_{1} (b_{1}) & \dots & σ_{1} (b_{n}) \\ ⋮ & ⋮ \\ σ_{n} (b_{1}) & \dots & σ_{n} (b_{n}) \end{matrix}) .

Then

Δ (B) = det (M)^{2} .

Proof ▼

By Lemma 2.2 we know that ${Tr}_{K / Q} (b_{i} b_{j}) = \sum_{k} σ_{k} (b_{i}) σ_{k} (b_{j})$ which is the same as the $(i, j)$ entry of $M^{t} M$ . Therefore

det (T_{B}) = det (M^{t} M) = det (M)^{2} .

Lemma 2.7 ✓

Let $K$ be a number field and $B = {1, α, α^{2}, \dots, α^{n - 1}}$ for some $α \in K$ . Then

Δ (B) = \prod_{i < j} (σ_{i} (α) - σ_{j} (α))^{2}

where $σ_{i}$ are the embeddings of $K$ into $C$ . Here $Δ (B)$ denotes the discriminant.

Proof ▼

First we recall a classical linear algebra result relating to the Vandermonde matrix, which states that

det (\begin{matrix} 1 & x_{1} & x_{1}^{2} & \dots & x_{1}^{n - 1} \\ ⋮ & ⋮ \\ 1 & x_{n} & x_{n}^{2} & \dots & x_{n}^{n - 1} \end{matrix}) = \prod_{i < j} (x_{i} - x_{j}) .

Combining this with Lemma 2.6 gives the result.

Lemma 2.8 ✓

Let $f$ be a monic irreducible polynomial over a number field $K$ and let $α$ be one of its roots in $C$ . Then

f^{'} (α) = \prod_{β \neq α} (α - β),

where the product is over the roots of $f$ different from $α$ .

Proof ▼

We first write $f (x) = (x - α) g (x)$ which we can do (over $C$ ) as $α$ is a root of $f$ , where now $g (x) = \prod_{β \neq α} (x - β)$ . Differentiating we get

f^{'} (x) = g (x) + (x - α) g^{'} (x) .

If we now evaluate at $α$ we get the result.

Lemma 2.9 ✓

Let $K = Q (α)$ be a number field with $n = [K : Q]$ and let $B = {1, α, α^{2}, \dots, α^{n - 1}}$ . Then

Δ (B) = (- 1)^{\frac{n (n - 1)}{2}} N_{K / Q} (m_{α}^{'} (α))

where $m_{α}^{'}$ is the derivative of $m_{α} (x)$ (which we recall denotes the minimal polynomial of $α$ ).

Proof ▼

By Lemma 2.7 we have $Δ (B) = \prod_{i < j} (α_{i} - α_{j})^{2}$ where $α_{k} := σ_{k} (α)$ . Next, we note that the number of terms in this product is $1 + 2 + \dots + (n - 1) = \frac{n (n - 1)}{2}$ . So if we write each term as $(α_{i} - α_{j})^{2} = - (α_{i} - α_{j}) (α_{j} - α_{i})$ we get

Δ (B) = (- 1)^{\frac{n (n - 1)}{2}} \prod_{i = 1}^{n} \prod_{i \neq j} (α_{i} - α_{j}) .

Now, by lemmas 2.8 and 2.1 we see that

N_{K / Q} (m_{α}^{'} (α)) = \prod_{i = 1}^{n} m_{α}^{'} (α_{i}) = \prod_{i = 1}^{n} \prod_{i \neq j} (α_{i} - α_{j}),

which gives the result.

Lemma 2.10 ✓

If $K$ is a number field and $α \in O_{K}$ then $N_{K / Q} (α)$ is in $Z$ .

Proof ▼

The proof is standard.

Lemma 2.11 ✓

If $K$ is a number field and $α \in O_{K}$ then ${Tr}_{K / Q} (α)$ is in $Z$ .

Proof ▼

The proof is standard.

Lemma 2.12 ✓

Let $K$ be a number field and $B = {b_{1}, \dots, b_{n}}$ be elements in $O_{K}$ , then $Δ (B) \in Z$ .

Proof ▼

Immediate by 2.11.

Lemma 2.13 ✓

Let $K = Q (α)$ be a number field, where $α$ is an algebraic integer. Let $B = {1, α, \dots, α^{[K : Q] - 1}}$ be the basis given by $α$ and let $x \in O_{K}$ . Then $Δ (B) x \in Z [α]$ .

Proof ▼

See the Lean proof.

Lemma 2.14 ✓

Let $K = Q (α)$ be a number field, where $α$ is an algebraic integer with minimal polynomial that is Eisenstein at $p$ . Let $x \in O_{K}$ such that $p^{n} x \in Z [α]$ for some $n$ . Then $x \in Z [α]$ .

Proof ▼

See the Lean proof. □