2 Discriminants of number fields

We recall basic facts about the discriminant.

Lemma 2.1 ✓

Let \(K\) be a number field, \(\alpha \in K\) and let \(\sigma _i\) be the embeddings of \(K\) into \(\mathbb {C}\). Then

\[ N_{K/\mathbb {Q}}(\alpha )=\prod _i \sigma _i(\alpha ) \]

Proof ▼

The proof is standard.

Lemma 2.2 ✓

Let \(K\) be a number field, \(\alpha \in K\) and let \(\sigma _i\) be the embeddings of \(K\) into \(\mathbb {C}\). Then

\[ \operatorname {Tr}_{K/\mathbb {Q}}(\alpha ) =\sum _i \sigma _i(\alpha ). \]

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

\[ \Delta (B)= \det \left(\begin{matrix} \operatorname {Tr}_{K/A}(b_1b_1) & \cdots & \operatorname {Tr}_{K/A}(b_1b_n) \\ \vdots & & \vdots \\ \operatorname {Tr}_{K/A}(b_nb_1) & \cdots & \operatorname {Tr}_{K/A}(b_nb_n) \end{matrix} \right). \]

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 \(\Delta (B) \neq 0\).

Proof ▼

The proof is standard.

Lemma 2.5 ✓

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

\[ \Delta (B)=\det (P)^2 \Delta (B'). \]

Proof ▼

The proof is standard.

Lemma 2.6 ✓

Let \(K\) be a number field with basis \(B=\{ b_1,\dots ,b_n\} \) and let \(\sigma _1,\dots ,\sigma _n\) be the embeddings of \(K\) into \(\mathbb {C}\). Now let \(M\) be the matrix

\[ \left(\begin{matrix} \sigma _1(b_1) & \cdots & \sigma _1(b_n) \\ \vdots & & \vdots \\ \sigma _n(b_1) & \cdots & \sigma _n(b_n) \end{matrix} \right). \]

Then

\[ \Delta (B)=\det (M)^2. \]

Proof ▼

By Lemma 2.2 we know that \(\operatorname {Tr}_{K/\mathbb {Q}}(b_i b_j)= \sum _k \sigma _k(b_i)\sigma _k(b_j)\) 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 \(B=\{ 1,\alpha ,\alpha ^2,\dots ,\alpha ^{n-1}\} \) for some \(\alpha \in K\). Then

\[ \Delta (B)=\prod _{i {\lt} j} (\sigma _i(\alpha )-\sigma _j(\alpha ))^2 \]

where \(\sigma _i\) are the embeddings of \(K \) into \(\mathbb {C}\). Here \(\Delta (B)\) denotes the discriminant.

Proof ▼

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

\[ \det \left(\begin{matrix} 1 & x_1 & x_1^2 & \cdots & x_1^{n-1} \\ \vdots & & & \vdots \\ 1 & x_n & x_n^2 & \cdots & x_n^{n-1} \end{matrix} \right) =\prod _{i{\lt}j} (x_i-x_j). \]

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 \(\alpha \) be one of its roots in \(\mathbb {C}\). Then

\[ f'(\alpha )=\prod _{\beta \neq \alpha } (\alpha -\beta ), \]

where the product is over the roots of \(f\) different from \(\alpha \).

Proof ▼

We first write \(f(x)=(x-\alpha )g(x)\) which we can do (over \(\mathbb {C}\)) as \(\alpha \) is a root of \(f\), where now \(g(x)=\prod _{\beta \neq \alpha } (x-\beta )\). Differentiating we get

\[ f'(x)=g(x)+(x-\alpha )g'(x). \]

If we now evaluate at \(\alpha \) we get the result.

Lemma 2.9 ✓

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

\[ \Delta (B)=(-1)^{\frac{n(n-1)}{2}}N_{K/\mathbb {Q}}(m_\alpha '(\alpha )) \]

where \(m_\alpha '\) is the derivative of \(m_\alpha (x)\) (which we recall denotes the minimal polynomial of \(\alpha \)).

Proof ▼

By Lemma 2.7 we have \(\Delta (B)=\prod _{i {\lt} j}(\alpha _i-\alpha _j)^2\) where \(\alpha _k:=\sigma _k(\alpha )\). Next, we note that the number of terms in this product is \(1+2+\cdots +(n-1)=\frac{n(n-1)}{2}\). So if we write each term as \((\alpha _i-\alpha _j)^2=-(\alpha _i-\alpha _j)(\alpha _j-\alpha _i)\) we get

\[ \Delta (B)=(-1)^{\frac{n(n-1)}{2}}\prod _{i=1}^n \prod _{i \neq j} (\alpha _i-\alpha _j). \]

Now, by lemmas 2.8 and 2.1 we see that

\[ N_{K/\mathbb {Q}}(m_\alpha '(\alpha ))=\prod _{i=1}^n m_\alpha '(\alpha _i)=\prod _{i=1}^n \prod _{i \neq j} (\alpha _i-\alpha _j), \]

which gives the result.

Lemma 2.10 ✓

If \(K\) is a number field and \(\alpha \in \mathcal{O}_K\) then \(N_{K/\mathbb {Q}}(\alpha )\) is in \(\mathbb {Z}\).

Proof ▼

The proof is standard.

Lemma 2.11 ✓

If \(K\) is a number field and \(\alpha \in \mathcal{O}_K\) then \(\operatorname {Tr}_{K/\mathbb {Q}}(\alpha )\) is in \(\mathbb {Z}\).

Proof ▼

The proof is standard.

Lemma 2.12 ✓

Let \(K\) be a number field and \(B=\{ b_1,\dots ,b_n\} \) be elements in \(\mathcal{O}_K\), then \(\Delta (B) \in \mathbb {Z}\).

Proof ▼

Immediate by 2.11.

Lemma 2.13 ✓

Let \(K = \mathbb {Q}(\alpha )\) be a number field, where \(\alpha \) is an algebraic integer. Let \(B = \{ 1, \alpha , \ldots , \alpha ^{[K : \mathbb {Q}] - 1} \} \) be the basis given by \(\alpha \) and let \(x \in \mathcal{O}_K\). Then \(\Delta (B)x \in \mathbb {Z}[\alpha ]\).

Proof ▼

See the Lean proof.

Lemma 2.14 ✓

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

Proof ▼

See the Lean proof.