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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the
field extension In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...
. The concept plays an important role in many parts of mathematics, including
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
and
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
—indeed in any area where fields appear prominently.


Definition and notation

Suppose that ''E''/''F'' is a
field extension In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...
. Then ''E'' may be considered as a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over ''F'' (the field of scalars). The
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
of this vector space is called the degree of the field extension, and it is denoted by 'E'':''F'' The degree may be finite or infinite, the field being called a finite extension or infinite extension accordingly. An extension ''E''/''F'' is also sometimes said to be simply finite if it is a finite extension; this should not be confused with the fields themselves being
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
s (fields with finitely many elements). The degree should not be confused with the
transcendence degree In mathematics, a transcendental extension L/K is a field extension such that there exists an element in the field L that is transcendental over the field K; that is, an element that is not a root of any univariate polynomial with coefficients ...
of a field; for example, the field Q(''X'') of
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s has infinite degree over Q, but transcendence degree only equal to 1.


The multiplicativity formula for degrees

Given three fields arranged in a
tower A tower is a tall Nonbuilding structure, structure, taller than it is wide, often by a significant factor. Towers are distinguished from guyed mast, masts by their lack of guy-wires and are therefore, along with tall buildings, self-supporting ...
, say ''K'' a subfield of ''L'' which is in turn a subfield of ''M'', there is a simple relation between the degrees of the three extensions ''L''/''K'', ''M''/''L'' and ''M''/''K'': : :K= :L\cdot :K In other words, the degree going from the "bottom" to the "top" field is just the product of the degrees going from the "bottom" to the "middle" and then from the "middle" to the "top". It is quite analogous to Lagrange's theorem in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, which relates the order of a group to the order and
index Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ...
of a subgroup — indeed
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
shows that this analogy is more than just a coincidence. The formula holds for both finite and infinite degree extensions. In the infinite case, the product is interpreted in the sense of products of
cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
s. In particular, this means that if ''M''/''K'' is finite, then both ''M''/''L'' and ''L''/''K'' are finite. If ''M''/''K'' is finite, then the formula imposes strong restrictions on the kinds of fields that can occur between ''M'' and ''K'', via simple arithmetical considerations. For example, if the degree 'M'':''K''is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
''p'', then for any intermediate field ''L'', one of two things can happen: either 'M'':''L''= ''p'' and 'L'':''K''= 1, in which case ''L'' is equal to ''K'', or 'M'':''L''= 1 and 'L'':''K''= ''p'', in which case ''L'' is equal to ''M''. Therefore, there are no intermediate fields (apart from ''M'' and ''K'' themselves).


Proof of the multiplicativity formula in the finite case

Suppose that ''K'', ''L'' and ''M'' form a tower of fields as in the degree formula above, and that both ''d'' = 'L'':''K''and ''e'' = 'M'':''L''are finite. This means that we may select a basis for ''L'' over ''K'', and a basis for ''M'' over ''L''. We will show that the elements ''u''''m''''w''''n'', for ''m'' ranging through 1, 2, ..., ''d'' and ''n'' ranging through 1, 2, ..., ''e'', form a basis for ''M''/''K''; since there are precisely ''de'' of them, this proves that the dimension of ''M''/''K'' is ''de'', which is the desired result. First we check that they span ''M''/''K''. If ''x'' is any element of ''M'', then since the ''w''''n'' form a basis for ''M'' over ''L'', we can find elements ''a''''n'' in ''L'' such that : x = \sum_^e a_n w_n = a_1 w_1 + \cdots + a_e w_e. Then, since the ''u''''m'' form a basis for ''L'' over ''K'', we can find elements ''b''''m'',''n'' in ''K'' such that for each ''n'', : a_n = \sum_^d b_ u_m = b_ u_1 + \cdots + b_ u_d. Then using the
distributive law In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary ...
and
associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...
of multiplication in ''M'' we have : x = \sum_^e \left(\sum_^d b_ u_m\right) w_n = \sum_^e \sum_^d b_ (u_m w_n), which shows that ''x'' is a linear combination of the ''u''''m''''w''''n'' with coefficients from ''K''; in other words they span ''M'' over ''K''. Secondly we must check that they are
linearly independent In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concep ...
over ''K''. So assume that : 0 = \sum_^e \sum_^d b_ (u_m w_n) for some coefficients ''b''''m'',''n'' in ''K''. Using distributivity and associativity again, we can group the terms as : 0 = \sum_^e \left(\sum_^d b_ u_m\right) w_n, and we see that the terms in parentheses must be zero, because they are elements of ''L'', and the ''w''''n'' are linearly independent over ''L''. That is, : 0 = \sum_^d b_ u_m for each ''n''. Then, since the ''b''''m'',''n'' coefficients are in ''K'', and the ''u''''m'' are linearly independent over ''K'', we must have that ''b''''m'',''n'' = 0 for all ''m'' and all ''n''. This shows that the elements ''u''''m''''w''''n'' are linearly independent over ''K''. This concludes the proof.


Proof of the formula in the infinite case

In this case, we start with bases ''u''α and ''w''β of ''L''/''K'' and ''M''/''L'' respectively, where α is taken from an indexing set ''A'', and β from an indexing set ''B''. Using an entirely similar argument as the one above, we find that the products ''u''α''w''β form a basis for ''M''/''K''. These are indexed by the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...
''A'' × ''B'', which by definition has
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
equal to the product of the cardinalities of ''A'' and ''B''.


Examples

* The
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s are a field extension over the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s with degree ''C:R= 2, and thus there are no non-trivial fields between them. * The field extension Q(, ), obtained by adjoining and to the field Q of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s, has degree 4, that is, ''Q(, ):Q= 4. The intermediate field Q() has degree 2 over Q; we conclude from the multiplicativity formula that ''Q(, ):Q()= 4/2 = 2. * The
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
(or Galois field) GF(125) = GF(53) has degree 3 over its subfield GF(5). More generally, if ''p'' is a prime and ''n'', ''m'' are positive integers with ''n'' dividing ''m'', then ''GF(''p''''m''):GF(''p''''n'')= ''m''/''n''. * The field extension C(''T'')/C, where C(''T'') is the field of
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s over C, has infinite degree (indeed it is a purely transcendental extension). This can be seen by observing that the elements 1, ''T'', ''T''2, etc., are linearly independent over C. * The field extension C(''T''2) also has infinite degree over C. However, if we view C(''T''2) as a subfield of C(''T''), then in fact ''C(''T''):C(''T''2)= 2. More generally, if ''X'' and ''Y'' are
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
s over a field ''K'', and ''F'' : ''X'' → ''Y'' is a surjective morphism between them of degree ''d'', then the function fields ''K''(''X'') and ''K''(''Y'') are both of infinite degree over ''K'', but the degree 'K''(''X''):''K''(''Y'')turns out to be equal to ''d''.


Generalization

Given two
division ring In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...
s ''E'' and ''F'' with ''F'' contained in ''E'' and the multiplication and addition of ''F'' being the restriction of the operations in ''E'', we can consider ''E'' as a vector space over ''F'' in two ways: having the scalars act on the left, giving a dimension 'E'':''F''sub>l, and having them act on the right, giving a dimension 'E'':''F''sub>r. The two dimensions need not agree. Both dimensions however satisfy a multiplication formula for towers of division rings; the proof above applies to left-acting scalars without change.


References

* page 215, Proof of the multiplicativity formula. * page 465, {{cite book , author=Jacobson, N. , authorlink=Nathan Jacobson, title=Basic Algebra II , publisher=W. H. Freeman and Company , year=1989 , isbn=0-7167-1933-9 Briefly discusses the infinite dimensional case. Field extensions