Quadratic Forms and Space-Time Block Codes from Generalized Quaternion and Biquaternion Algebras Thomas Unger and Nadya Markin

Home , Biquaternion

1 Quadratic Forms and Space-Time Block Codes from Generalized Quaternion and Biquaternion Algebras Thomas Unger and Nadya Markin

Abstract—In the context of space-time block codes (STBCs), construction will always work with F replaced by K. How- the theory of generalized quaternion and biquaternion algebras ever, when D is noncommutative, one should consider D as a (i.e., tensor products of two quaternion algebras) over arbitrary right K-vector space, i.e., elements of D should be multiplied base fields is presented, as well as quadratic form theoretic criteria to check if such algebras are division algebras. For on the right by scalars from K. It should be taken into account 2 base fields relevant to STBCs, these criteria are exploited, via that in both cases dimF D = (dimK D) , resulting in K- Springer’s theorem, to construct several explicit infinite families matrices of smaller size. of (bi-)quaternion division algebras. These can be used to obtain We are only interested in noncommutative division algebras 2 × 2 and 4 × 4 STBCs via the left regular representation. in this paper. In the STBC literature two large classes of Index Terms—Space-time block codes, division algebras, quad- algebras have been studied: cyclic algebras (e.g. in [2]) and ratic forms. crossed-product algebras (e.g. in [3] and [4]). The latter are characterized by requiring that K is a strictly maximal subfield D F I.INTRODUCTION of which is a Galois extension of . Cyclic algebras are crossed-product algebras with cyclic Galois group Gal(K/F ). VER since Alamouti’s work [1] a decade ago, division We refer to [5] for a historical survey of such algebras. E algebras – both commutative (i.e., fields) and noncom- Determining when precisely these algebras are division mutative – have played an increasingly important role in the algebras can be a nontrivial problem. In this paper we discuss construction of space-time block codes (STBCs) for wireless two classes of algebras, namely (generalized) quaternion and communication. biquaternion algebras. They have characterizations in terms Precisely how this is done is explained very thoroughly in of quadratic forms, a fact that has not fully been exploited [2]. From a mathematical point of view, one wants to construct, yet in the coding theory literature. There are particularly nice for a given integer n, a set S ⊂ Mn(F ) of n×n-matrices over quadratic form-theoretic criteria to determine when they are some field F such that the difference of any two elements of division algebras. S has nonzero determinant. Although many examples of quaternion algebras have ap- The field F is usually an extension of the field Q of rational peared in the literature (first and foremost the Alamouti code numbers, containing an appropriate signal constellation. The of course), they often do so disguised as cyclic algebras (of set S ⊂ Mn(F ) can be obtained as follows: choose some which they are special cases), which hides their quadratic division algebra D of dimension n over F (i.e., D is simul- form-theoretic characterization. taneously an F -vector space and a ring) and consider the left Biquaternion algebras are tensor products of two quaternion regular representation algebras (in the older literature, the term also refers to Hamil- λ : D → End (D), a 7→ λ , ton’s quaternions considered over the field of complex num- F a bers). They are crossed-product algebras with Galois group the where λa is left multiplication by a, λa(x) = ax for all x ∈ Klein 4-group (but the converse inclusion is not true), and are D. Here EndF (D) is the algebra of F -linear transformations thus special cases of the algebras in [4]. Biquaternion algebras on D. After a choice of F -basis for D, EndF (D) can be can also be characterized by a quadratic form. In particular it identified with the algebra Mn(F ). In this way we get an is relatively straightforward to determine when such an algebra embedding λ : D,→ Mn(F ). Since all nonzero elements is a division algebra (at least over base fields that are relevant of D are invertible, all nonzero matrices in λ(D) will have for STBCs). It is also easy to produce biquaternion division nonzero determinant. The set S can then be chosen to be (a algebras as tensor products of quaternion division algebras, subset of) λ(D). The difference of any two elements of S will addressing an issue raised in [3, p. 3926]. thus have nonzero determinant. Quaternion division algebras can be used to construct 2 × 2 There are some caveats here: for coding purposes it is often STBCs and biquaternion division algebras to construct 4 × 4 more convenient to consider D as a vector space over some STBCs. subfield K, maximal with respect to inclusion (F ⊂ K ⊂ D). This paper is organized as follows: in Section II we present a When D is commutative (in other words a field), the above brief introduction to quadratic forms. In Sections III and IV we respectively present the theory of quaternion and biquaternion T. Unger and N. Markin are with the School of Mathematical Sci- algebras over arbitrary fields. We also give general quadratic ences, University College Dublin, Belfield, Dublin 4, Ireland (e-mail: [email protected] and [email protected]). form theoretic criteria for such algebras to be division algebras. Version of July 1, 2008 In Section V we obtain matrix representations of quaternion 2

n and biquaternion algebras over arbitrary fields by means of for all v = (v1, . . . , vn) ∈ F and scalars a1, . . . , an ∈ F . It the left regular representation. is convenient to employ the short-hand notation Then we shift our attention to fields that are actually used ψ = ha , . . . , a i in space-time block coding: number fields and their transcen- 1 n dental extensions. Such fields can be embedded into bigger to indicate that the matrix of ψ has diagonal entries a1, . . . , an fields that are complete with respect to a discrete valuation, and zeroes everywhere else. Thus we can write or CDV fields for short. For CDV fields the quadratic form ϕ ' ha , . . . , a i. theoretic criterium for a (bi-)quaternion algebra to be division 1 n can be tested in a finite number of steps by means of Springer’s It is customary to consider quadratic forms up to isometry. As theorem (Section VI). a consequence the entries in ha1, . . . , ani may be permuted In the final two sections we give examples of infinite fam- and changed by nonzero squares (i.e., any ai may be replaced 2 × ilies of non-isomorphic quaternion and biquaternion algebras by α ai for any α ∈ F ). over suitable fields, exploiting Springer’s theorem. Together The dimension of ϕ is the integer n (i.e., the dimension of with the matrix representations from Section V these can be the underlying vector space). If ϕ0 is another quadratic form 0 used for the explicit construction of 2 × 2 and 4 × 4 STBCs. over F , say of dimension m, ϕ ' hb1, . . . , bmi, then the The Appendices contains preliminaries from discrete valu- orthogonal sum of ϕ and ϕ0, defined by ation theory and number theory. 0 ϕ ⊥ ϕ := ha1, . . . , an, b1, . . . , bmi n+m II.QUADRATIC FORMS is a quadratic form on F which is uniquely defined up to isometry. For the convenience of the reader we recall some basic facts The quadratic forms ϕ in this paper will always be from quadratic form theory. We refer to [6] for the details. Let nonsingular in the sense that their determinant det ϕ := F be a field and write F × := F \{0}. We make the technical a a ··· a ∈ F/F 2 is always nonzero. assumption that the characteristic of F is different from 2 (in 1 2 n A quadratic form ϕ on V is called isotropic over F if other words that 2 is invertible in F ). The fields considered there exists a nonzero vector v ∈ V such that ϕ(v) = 0 in this paper are all extensions of the rational numbers and Q and anisotropic over F otherwise. thus will satisfy this condition (but the finite field does not, F2 Note that an anisotropic form over F may become isotropic for example). after scalar extension to a bigger field. Let V be a finite dimensional F -vector space. Remark 2.2: Let ϕ = ha , . . . , a i. If there are Definition 2.1: A quadratic form on V (over F ) is a map 1 n b , b , . . . , b ∈ F × such that ϕ ' h1, −1, b , b , . . . , b i, then ϕ : V → F satisfying the following two properties: 3 4 n 3 4 n ϕ is clearly isotropic. The converse is also true. (i) ϕ(αv) = α2ϕ(v) for all v ∈ V and all α ∈ F ; (ii) the map bϕ : V × V → F defined by III.QUATERNION ALGEBRAS × bϕ(v, w) := ϕ(v + w) − ϕ(v) − ϕ(w) Definition 3.1: Let a, b ∈ F .A (generalized) quaternion algebra over F is a 4-dimensional F -vector space with F - for all v, w ∈ V is bilinear (and automatically symmet- basis B = {1, i, j, k}, whose ring structure is determined by ric). 2 2 n i = a, j = b and ij = −ji = k. (2) Concretely, if we identify V with F , where n = dimF V , then ϕ is just a homogeneous polynomial of degree two: for We denote this algebra by (a, b)F . Its elements are called 1 ≤ i, j ≤ n there exist scalars aij ∈ F such that quaternions. 2 n It follows from the defining relations (2) that k = −ab and X ϕ(v) = ϕ(v1, . . . , vn) = aijvivj, (1) that i, j and k anti-commute. Addition of quaternions and i,j=1 multiplication of a quaternion by a scalar is component wise, while the product of two quaternions reduces to computing for all v = (v , . . . , v ) ∈ F n. 1 n sums of scalar multiples of products of the form ξη with ξ, If we let A = (a ) ∈ M (F ) we can write (1) in ij 1≤i,j≤n n η ∈ B. The defining relations (2) should be used to simplify matrix form as these products. Note that the product of two quaternions is not ϕ(v) = vAvt, commutative. where t denotes transposition. Example 3.2: Hamilton’s quaternions form the algebra H = If ψ is another quadratic form on V , with matrix B say, (−1, −1)R. then we call ϕ and ψ isometric if the matrices A and B are Let q = q0 + q1i + q2j + q3k ∈ Q := (a, b)F (with congruent, i.e., if there exists an invertible matrix C such that q0, q1, q2, q3 ∈ F ). The conjugate of q, denoted q, is defined B = CACt. In this case we write ϕ ' ψ. by One can show that a quadratic form ϕ is always isometric q := q0 − q1i − q2j − q3k. to a quadratic form ψ whose matrix is diagonal, The norm map N : Q → F is defined by N(q) := qq for all n X q ∈ Q. A straightforward computation shows that ψ(v) = a v2 i i 2 2 2 2 i=1 N(q) := qq = qq = q0 − aq1 − bq2 + abq3. (3) 3

Thus we may regard N as a quadratic form in four variables Deﬁnition 4.2: Let B := (a, b)F ⊗F (c, d)F . The six- q0, q1, q2, q3, called the norm form of Q. Using the notation dimensional quadratic form from Section II we write ϕ = ha, b, −ab − c, −d, cdi N = h1, −a, −b, abi. over F is called the Albert form of B. Theorem 3.3: Q = (a, b) The quaternion algebra F is a divi- This form is named after Adrian A. Albert who carried out N = h1, −a, −b, abi sion algebra if and only if its norm form much of the pioneering work on the structure of algebras in F is anisotropic over . the 1930s. Proof: ⇒ If N(q) = qq were equal to zero for Theorem 4.3: The biquaternion algebra B = (a, b) ⊗ some nonzero q ∈ Q, then Q would contain zero divisors, F F (c, d) is a division algebra if and only if its Albert form contradicting the assumption that Q is a division algebra. F ϕ = ha, b, −ab, −c, −d, cdi is anisotropic over F . ⇐ Let q be any nonzero element of Q. From qq = qq = The proof of this theorem is beyond the scope of this paper. N(q) 6= 0 we deduce that q is invertible with q−1 = q/N(q). Remark 4.4: If the Albert form ϕ of B is isotropic over F , then one can show that B isomorphic to the full matrix algebra Example 3.4: Hamilton’s quaternion algebra M (F ) if and only if ϕ is isometric to h1, −1, 1, −1, 1, −1i = (−1, −1) is a division algebra since its norm 4 H R and also that B =∼ M (D) for some quaternion division form h1, 1, 1, 1i is clearly anisotropic over . The quaternion 2 R algebra D if and only if ϕ ' h1, −1, e, f, g, hi with he, f, g, hi algebra (−1, 1) is not a division algebra since its norm form R anisotropic. h1, 1, −1, −1i is isotropic over . R Remark 4.5: One can show that two biquaternion algebras Remark 3.5: Let Q = (a, b) . One can show that if Q is F over F are isomorphic as F -algebras if and only if their Albert not a division algebra, that it is then necessarily isomorphic forms are similar, i.e., isometric up to a multiplicative constant to the full matrix algebra M (F ). This thus happens precisely 2 from F . when the norm form of Q is isotropic. Remark 3.6: One can show that two quaternion algebras over F are isomorphic as F -algebras if and only if their norm V. MATRIX REPRESENTATIONS forms are isometric quadratic forms. A. The Quaternion Case

IV. BIQUATERNION ALGEBRAS Let Q = (a, b)F be a quaternion division algebra with F - basis {1, i, j, k}, so that i2 = a and j2 = b. The field K := Definition 4.1: A biquaternion algebra over F is a tensor √ F ( a) = F (i) is a maximal subfield of Q with Galois group product of two quaternion algebras over F , Gal(K/F ) = {1, σ} where σ (the nontrivial√ automorphism√ of (a, b)F ⊗F (c, d)F , K that fixes F ) is determined by σ( a) = − a. Let x = q + q i + q j + q k ∈ Q (with q , q , q , q ∈ F ). a, b, c, d ∈ F × biquaternions 0 1 2 3 0 1 2 3 where . Its elements are called . Since Let B1 := {1, i1, j1, k1} and B2 := {1, i2, j2, k2} be F - bases for (a, b)F and (c, d)F , respectively. The elements of x = q0 +q1i+q2j+q3k = (q0 +q1i)+j(q2 −q3i) = x0 +jx1 B1 are assumed to commute with the elements of B2. The ordered set with x0 = q0 + q1i and x1 = q2 − q3i in K, we can write

B := {ξ ⊗ η | ξ ∈ B1, η ∈ B2} Q = F (i) ⊕ jF (i) = K ⊕ jK. is an F -basis for B := (a, b)F ⊗F (c, d)F . Thus dimF B = 16 Hence Q can be considered as a two-dimensional right K- and we can express a biquaternion x ∈ B uniquely in the form vector space with K-basis {1, j}. We thus multiply elements X X x from Q with scalars α from K on the right to get xα. In x = xξ,η ξ ⊗ η, computations we may bump into expressions of the form αx ξ∈B1 η∈B2 with scalars on the left. They can be brought into the correct with xξ,η ∈ F . Addition of biquaternions and multiplication form by means of the “hopping over” rule of a biquaternion by a scalar is component wise, while the product of two biquaternions reduces to computing sums of αx = xσ(α) (4) scalar multiples of products of the form (which can be veriﬁed to be true by direct computation). 0 0 0 0 (ξ ⊗ η)(ξ ⊗ η ) = ξξ ⊗ ηη , Theorem 5.1: 0 0 ( ) where ξ, ξ ∈ B1 and η, η ∈ B2. The deﬁning relations (2) ∼ x0 bσ(x1) should be used to simplify these products. For example, Q = x0, x1 ∈ K x1 σ(x0) 2 2 (i1 ⊗ j2)(i1 ⊗ k2) = i1 ⊗ j2k2 = a ⊗ j2i2j2 = a ⊗ (−i2j2) Proof: Let λ : Q,→ EndK (Q), x 7→ λx be the left = a ⊗ (−di2) = −ad(1 ⊗ i2). regular representation. We can identify EndK (Q) with M2(K) Note that the product of two biquaternions is not commutative. by mapping λx to its matrix with respect to the K-basis {1, j} 4

of Q, which we will denote by X. Let x = x0 + jx1 ∈ Q. Theorem 5.5: Since   x0 bσ(x1) dτ(x2) bdστ(x12)   λx(1) = x = x0 + jx1,  x1 σ(x0) dτ(x12) dστ(x2)  B =∼   , 2  x2 bσ(x12) τ(x0) bστ(x1)  λx(j) = xj = (x0 + jx1)j = x0j + jx1j = jσ(x0) + j σ(x1)    x12 σ(x2) τ(x1) στ(x0)  = bσ(x1) + jσ(x0) where x , x , x , x ∈ K. (where we used the “hopping over” rule (4) and the fact that 0 1 2 12 The proof is similar to the proof of Theorem 5.1 but this j2 = b), we obtain time the “hopping over” rules (5) should be used. x bσ(x ) X = 0 1 . x σ(x ) 1 0 VI.SPRINGER’S THEOREM Since the map x 7→ X yields an isomorphism of Q with its So far we have described quaternion and biquaternion alge- image inside M2(K) we are done. bras over arbitrary fields (in which 2 is invertible) and shown Example 5.2: Letting F = R, K = C and Q = (−1, −1)R how to compute their matrix representations. In addition we we obtain the Alamouti code. have seen that a given quaternion or biquaternion algebra is a More examples will be constructed in Section VII. division algebra if and only if its norm form or Albert form, Remark 5.3: Note that respectively, is anisotropic. The next question is how to determine when a quadratic det X = x σ(x )−bx σ(x ) = q2−aq2−bq2+abq2 = N(x), 0 0 1 1 0 1 2 3 form over a given field is actually anisotropic. In general this is confirming once again that invertible elements of Q (i.e., ele- a difficult problem, but for the fields in which we are interested ments with nonzero norm) correspond to invertible matrices. we are in luck! The idea is to embed them into bigger fields √ Remark 5.4: The field K = F ( a) is a so-called splitting that are complete with respect to a discrete valuation. For such ∼ fields there exists an easy criterion for anisotropy: Springer’s field for Q in the sense that Q ⊗F K = M2(K). theorem. If the quadratic form is anisotropic over the bigger field, then it was already anisotropic over the smaller field. B. The Biquaternion Case Applying this to a norm form or Albert form we get that if a quaternion or biquaternion algebra is a division algebra over Let B := (a, b)F ⊗F (c, d)F be a biquaternion division 2 the bigger field, then it was already division over the smaller algebra. Let B1, B2 and B be as in Section IV. Then i1 = a, 2 2 2 √ √ field. j1 = b, i2 = c and j2 = d. The field K := F ( a, c) is a maximal subfield of B with Galois group Gal(K/F ) = We will now present the details of this approach. Our fields √ √ of interest are: {1, σ, τ,√ στ} where√ σ and τ are determined by σ( a) = − a and τ( c) = − c, respectively, and where στ denotes the • number fields F (i.e., finite extensions of the field Q of composition of σ and τ. rational numbers); The field K is a biquadratic extension of the field F . • rational function field extensions of F of the form F (x, y). The√ following√ diagram illustrates the relationship between K, F ( a), F ( c) and F : These are discretely valued fields. We embed them into bigger √ √ fields: K = F ( a, c) • F,→ Fp: p-adic field; oo OOO 2 ooo OOO2 • F (x, y) ,→ F ((x))((y)): iterated Laurent series field. ooo OOO √ ooo OO √ These bigger fields are complete with respect to a discrete F ( a) F ( c) valuation (or CDV fields for short). They are the completions OO o OOO ooo of F and F (x, y), respectively, for an appropriate discrete OOO ooo σ OOO ooo τ valuation. We refer to Appendix A for more background. OO ooo F Theorem 6.1 (Springer’s theorem on CDV fields): Let F be a CDV field with residue field F , uniformizer π and group A K-basis for B is given by {1 ⊗ 1, 1 ⊗ j , j ⊗ 1, j ⊗ j }. 2 1 1 2 of units U . Assume that F is non-dyadic (i.e., char(F ) 6= 2). One can easily verify that B is a right K-vector space which Let ϕ be a quadratic form over F . Write can be identified with

ϕ = ϕ1 ⊥ hπiϕ2 K ⊕ j1K ⊕ j2K ⊕ j1j2K.

where ϕ1 = hu1, . . . , uri, ϕ2 = hur+1, . . . , uni and ui ∈ U This time there are several “hopping over” rules for multipli- for 1 ≤ i ≤ n. Then ϕ is anisotropic over F if and only if cation by scalars on the left, determined by the “ﬁrst residue form” ϕ1 = hu1,..., uri and the “second residue form” ϕ = hu ,..., u i are anisotropic over F . αj1 = j1σ(α), αj2 = j2τ(α) and αj1j2 = j1j2στ(α) (5) 2 r+1 n We refer to [6, Chapter VI, §1] for more information on for all α ∈ K. Springer’s theorem. 5

VII. 2 × 2 STBCSFROM QUATERNION ALGEBRAS can be represented by the elements {0, 1, −1, i, −i}. It follows [i]/(2 + i) 0, 1, −1 Using Springer’s theorem we can construct infinite families that the squares in Z are precisely . By our b of quaternion division algebras over number fields. They will choice of and Theorem 7.1 we are done. give rise to 2×2 STBCs by means of the matrix representation In the next example we pick a prime ideal p of Z[i] with a in Theorem 5.1. We refer to Appendix B for more background non-trivial inertial degree in order to construct another infinite on number fields. family of quaternion division algebras over Q(i). Example 7.5: p = 7 [i] b = a ai a Theorem 7.1: Let F be a number field, p a prime ideal of Let Z . Let or , where is an integer such that a ≡ 3, 5 or 6 mod 7. Then (7, b) (i) is F and π an element of F such that vp(π) = 1. Then for any Q element b ∈ F which is not a square modulo p, the quaternion a division algebra. algebra (π, b) is a division algebra. Proof: By Theorem B.3, the ideal p = 7Z[i] is the F 7 7 Proof: We show that the norm form N = h1, −π, −b, πbi principal ideal generated by and its inertial degree over Z 2 [i]/p 72 = 49 is anisotropic over F . Consider the p-adic valuation on F . is . Hence the number of elements in Z is . As an additive group the residue field Z[i]/p is isomorphic Let Op denote the valuation ring. We obtain the following to Z/7Z ⊕ Z/7Z: it has 49 elements of the form a + bi with decomposition of N in Fp: a, b ∈ Z/7Z. In particular, we can verify that our choices N = ϕ1 ⊥ hπiϕ2, of b are among non-square elements of this field. Again, by Theorem 7.1 we are done. where ϕ1 = h1, −bi and ϕ2 = h−1, bi. Let ϕi denote the Note that there are non-squares in the residue field of more reduction of ϕi modulo p. Since we chose b to be a non-square general form. More precisely, there are 25 squares in [i]/p, 2 2 Z in OF /p, the equation x b = y has no non-trivial solutions in which all have the form (x + yi)2 = (x2 − y2) + i2xy, where /p, therefore both ϕ and ϕ are anisotropic in /p. By OF 1 2 OF x, y ∈ Z and the bar denotes reduction modulo 7. This leaves Springer’s theorem it follows that N is anisotropic over Fp. 24 choices for the congruence class of b which will satisfy Since F is embedded in Fp, any non-trivial zero of N over F the condition of Theorem 7.1. Picking an ideal with non- would give rise to a non-trivial zero over Fp. It follows that trivial inertial degree gives rise to candidates b which cannot N is anisotropic over F . Hence (π, b) is a division algebra. F be obtained from the field Z/7Z. The previous two examples are relevant for QAM signal Corollary 7.2: Let F be a number field. There are infinitely constellations. The next example is relevant for PSK signal many quaternion division algebras (π, b)F . constellations. Proof: For any number field F , there are infinitely many Example 7.6: Let F = Q(ζ3), where ζ3 is a primitive third primes p in OF . Since any element of OF is contained in only root of√ unity. Let b ∈ Z such that b = 3, 5, or 6 mod 7. Then finitely many primes, this gives rise to infinitely many elements (2 + −3, b)F is a division algebra. π = πp. Having picked π, we can always find b ∈ OF which Proof: Again, we demonstrate two things: the fact that 2 √ is non-square modulo π. This is because the map x 7→ x in 2 + −3 has p-adic valuation 1 for a suitable ideal p and OF /p is never surjective (since p is not a prime above 2, it the fact that 3, 5, 6 are non-squares modulo p. Consider the has a non-trivial kernel containing −1). decomposition of the ideal 7OF in OF . By Theorem B.2, the In the examples below we illustrate how to use Theorem 7.1. inertial degree f of the ideal (7) in OF is 1, since 7 ≡ 1 We start with the simplest case when the number field F is mod 3. Indeed, we see that there are two distinct prime√ ideals . Q (i.e.,√ r = 2) in the decomposition√ of 7OF = (2 + −3)(2 − Example 7.3: The quaternion algebra (3, 5) is a division ∼ Q −3). Therefore OF /(2+ −3) =√Z/7Z. We can verify that algebra. 3, 5, 6 are not squares in OF /(2 + −3). Hence we are done Proof: The ideal p in this case is simply 3Z, with by Theorem 7.1. uniformizer 3. Let x denote the reduction of x modulo p. The squares in /3 are {0, 1}. Since 5 ≡ 2 mod 3, it is not a Z Z VIII. 4 × 4 STBCSFROM BIQUATERNION ALGEBRAS square, hence the conditions of Theorem 7.1 are satisfied. We The following theorem can be used to construct 4×4 STBCs conclude that (3, 5)Q is a division algebra. Note that instead of 5 we could use any element congruent by means of the matrix representation in Theorem 5.1. to 2 modulo 3. Theorem 8.1: Let K = F (x, y), where F is a number field, and x, y are transcendental over F . Let B = (a, x) ⊗(b, y) . In the next example we construct an infinite√ family of K K a, b ∈ F × a, b, ab 6∈ F ×2 B quaternion division algebras over Q(i), where i = −1. If are such that , then is a division algebra over K. Example 7.4: Let p = (2 + i)Z[i] and let b ≡ ±i mod p. Proof: The biquaternion algebra B will be a division Then (2 + i, b) (i) is a division algebra. Q algebra over K if and only if its Albert form Proof: Note that 2 + i is prime in Z[i] since its norm, 5, is prime in Z. Hence p is a prime ideal of Z[i]. We need ϕ = ha, x, −ax, −b, −y, byi to show that 2 + i is not a square in Z[i]/p. Note that p lies above the prime ideal (5) = 5Z: by Theorem B.3 we have is anisotropic over K. We will show that this is the case by (5)Z[i] = (2 + i)(2 − i) = pp. In addition the inertial degree an iterated application of Springer’s Theorem. of p over (5) is equal to 1. Hence the residue field Z[i]/p Consider the embedding of K = F (x, y) into the iterated is a trivial extension of Z/5Z. Thus Z[i]/p has order 5 and Laurent series field F ((x))((y)). (Note that F ((x))((y)) 6= 6

F ((y))((x)).). Let vy denote the valuation of F ((x))((y)) at Example A.1: Consider a prime ideal pZ of Z. Any nonzero the ideal (y). We obtain the decomposition rational number can be expressed as pax/y, where a, x and y are integers and p - xy. Define the p-adic valuation on Q by ϕ = ϕ1 ⊥ hyiϕ2, a vp : p x/y 7→ a. Then where ϕ = ha, x, −ax, −bi and ϕ = h−1, bi. The form ϕ ( ) 1 2 x will be anisotropic over F ((x))((y)) if and only if the forms Ovp = ∈ Q p - y , ∼ y ϕ1 and ϕ2 are anisotropic over F ((x))((y))/(y) = F ((x)), ( ) the residue field of vy. The form ϕ2 is clearly anisotropic x over F ((x)) by our assumption. We can decompose the form mvp = ∈ p y, p | x , and y Q - ϕ as 1 ( ) ϕ1 = ψ1 ⊥ hxiψ2, x Uv = ∈ p xy . p y Q - where ψ1 = ha, −bi and ψ2 = h1, −ai. By our assumption the residue forms ψ and ψ are clearly anisotropic over 1 2 Qp denotes the completion of Q with respect to the p-adic F ((x))/(x) =∼ F so that ϕ is anisotropic over F ((x)). In 1 valuation and Zp denotes the ring of integers of Qp. The conclusion, the form ϕ is anisotropic over F ((x))((y)), and ∼ ∼ residue field Qp = Zp/pZp = Fp is a finite field of order thus certainly anisotropic over the smaller field K. p. Remark 8.2: Note that picking transcendental elements of Similarly we can define p-adic valuations on number fields, iα iβ the form x = e , y = e guarantees that they have Euclidean as in the next example. norm 1. In the context of STBCs this can be used to satisfy Example A.2: Let F be a number field with ring of integers uniform energy constraints. OF . Any prime ideal p of OF induces a discrete valuation on F . Namely, let vp map a nonzero element x of OF to the i APPENDIX A largest integer i, such that x ∈ p , and let vp(0) = ∞. Extend PRELIMINARIESFROM DISCRETE VALUATION THEORY this map to F by letting vp(x/y) := vp(x) − vp(y). The map We refer to [7, Chapter 1] for the details. Let F be a field. just defined is a discrete valuation on F . We let Fp denote A discrete valuation on F is a map Fvp , the completion of F under vp. Similarly let Op and Up denote the ring of integers of Fp (also called p-adic integers) v : F → Z ∪ {∞} and the group of units of Op (called p-adic units), respectively. Example A.3: Let F be a field and let K = F (x) . For with the following properties: N • v(a) = ∞ ⇔ a = 0, X i • v(ab) = v(a) + v(b), K 3 f(x) = aix , am 6= 0, m ∈ Z • v(a + b) ≥ min(v(a), v(b)). i=m

Given a valuation v on F , let define vx(f) = m. Then Ovx = F [x], with residue field F . The completion of K under vx is the ring of Laurent series, Ov := {a ∈ F | v(a) ≥ 0}, P∞ i denoted by F ((x)). Its elements are of the form i=m aix mv := {a ∈ F | v(a) > 0}, and with m ∈ Z. The ring of integers of F ((x)) is the ring of F [[x]] Uv := {a ∈ F | v(a) = 0}. formal power series , whose elements are of the form P∞ i i=0 aix . The objects just defined are the valuation ring, the maximal ideal and the group of units of v, respectively. The ring Ov is APPENDIX B a local ring and mv is its unique maximal ideal. The residue PRELIMINARIESFROM NUMBER THEORY field of v, F v, is defined to be Ov/mv. Any discrete valuation v on F induces an absolute value Next we look at how prime ideals behave in extensions of |a| := cv(a), where 0 < c < 1 is a real constant. Two absolute number fields. Let F be a number field, i.e., a finite extension values | · |1, | · |2 are said to be equivalent if there is a constant of Q. Let K be a finite extension of F . If p is a prime ideal t p t such that |a|1 = |a|2 for all a ∈ F . Let | · |v denote the of OF , then OK is an ideal of OK with unique factorization equivalence class of absolute values induced by v. If v is a into prime ideals of OK : discrete valuation on F , then F has a completion Fv with e1 er pOK = P1 ··· Pr . respect to |·|v. This completion is unique up to an isomorphism over F . An element π of Fv such that v(π) = 1 is called a The integer ei is called the ramification index of Pi over p. uniformizer. Any two uniformizers differ by multiplication by The inertial degree fi is defined to be [OK /Pi : OF /p], the × i a unit. Any element x ∈ Fv has the form x = π u, where i degree of the field extension OK /Pi of OF /p. is an integer and u ∈ Uv is a unit. This gives us the following When K/F is a Galois extension, the ramification indices × isomorphism of the multiplicative group of Fv : (resp. inertial degrees) are the same for all i and we denote × ∼ them simply by ep (resp. fp). The following theorem describes Fv = Z × Uv. the relationship between inertial and ramification degrees in Below we give some examples of discrete valuations. this case. 7

Theorem B.1 ( [8, Theorem 21]): Let K be a Galois extension of F of degree n and let p be a prime ideal of F . Then ep pOK = (P1 ··· Prp ) with rpepfp = n = [K : F ]. When the prime ideal in the ground ﬁeld F is understood, we may drop the indices and refer to these parameters as r, e, f. In the special case of cyclotomic extensions of Q we have the following result: Theorem B.2 ( [8, Theorem 26]): Let F = Q(ζm), where ζm is a primitive m-th root of unity. Let p be a prime not dividing m. Then the inertial degree fp in F is the multiplicative order of p modulo m. Finally, for the decomposition of primes in the extension Q(i) of Q (with i2 = −1) we obtain the following theorem as a special case of [8, Theorem 25]: Theorem B.3: Let F = Q(i) (and thus OF = Z[i]) and let p be an odd prime. If p ≡ 1 mod 4 then

pZ[i] = (π)(π) = pp, where π and π are non-associated, prime and of norm p in Z[i]. Thus rp = 2, fp = ep = 1. If p ≡ 3 mod 4 then

pZ[i] = (p), i.e., the ideal pZ remains inert in Z[i] and rp = ep = 1, fp = 2.

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