what is galois field?
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http://www.phys.ufl.edu/docs/matlab/toolbox/comm/tutor3.html
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I'll assume you want n to be a positive integer. Consider F[a], the field obtained by adjoining a. Then a^n is in F[a] and the index of F[a] over F is finite. But every element of a field of finite index over F will be algebraic over F.
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As a^2 is contained in F(a), clearly F(a^2) is contained in F(a). So we need to show the reverse inclusion.\n\nWe have the chain of fields F < F(a^2) <= F(a).\n\n(where I use <= and < for subset and proper subset containment) \n\nand hence [F(a) : F] = [F(a) : F(a^2)][F(a^2) : F]. Set n = [F(a) : F] and m = [F(a^2) : F]. The preceeding equation implies that m divides n. It will suffice to that m = n. Let f(X) = X^m + c_{m-1}X^{m-1} + ... + c_1X + c_0 denote the minimal polynomial of a^2 over F. Then f(X^2) = X^{2m} + c_{m-1}X^{2(m-1)} + ... + c_1X^2 + c_0 has a as a root. This implies that the minimal polynomial of a has degree less than or equal to 2m. So we get m divides n and n <= 2m. The only ways for this to occur are when m=n or when 2m = n. But n is odd, so n does not equal 2m. Hence m = n. This completes the proof.
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The Ring of polynoms over the Field lets say G2 (just for denoting) is the right algebraic model to use if we want to study CRCs. \n\nA sequence of bit like 10011101 is modeled by 1x7 + 0x6 + 0x5 + 1x4 + 1x3 + 1x2 + 0x1 + 1x0 which we can write shorter as x7 + x4 + x3 + x2 + 1.\n\nI hope I helped...
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Kroydron or something like that. \nHere is a place you can look up value \nhttp://valueguide.pga.com/index.php
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Googolplex is the number .\n\nIt can also be written as 10googol, or as a 1 followed by a googol (10 to the 100th power) zeroes. Note that  is the same as  because exponentiation works from the top down. The term googol was coined by Milton Sirotta, nephew of mathematician Edward Kasner. Googolplex was coined by Kasner to define an especially large number by extension from his nephew's idea.
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An element of an adèle group, sometimes called a repartition in older literature (e.g., Chevalley 1951, p. 25). Adèles arise in both number fields and function fields. The adèles of a number field are the additive subgroups of all elements in , where  is the field place, whose absolute value is  at all but finitely many s. \n\nLet  be a function field of algebraic functions of one variable. Then a map  which assigns to every field place  of  an element  of  such that there are only a finite number of field places  for which  is called an adèle (Chevalley 1951, p. 1951).
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ALGO\n\nAlgo is an algebraie programmed language.\n\nClick on the URL below for additional information concerning ALGO..\n\nen.wikipedia.org/wiki/ALGO - 13k
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In mathematics, a polynomial is an expression in which a finite number of constants and variables are combined using only addition, subtraction, multiplication, and non-negative whole number exponents (raising to a power).
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Nodes for important fields, attributes for subfields.
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In calculated query field:\n============================\nTotal: [Field1]*[Field2]\n\nIn a form field:\n===============\n=[Field1]*[Field2]
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A ring and a field both have two operations, usually denoted by * and +. For both, associativity of both * and + are assumed. It is assumed that + has an identity, 0, and inverses, x|-> -x and that * is distributive over +.\n\nUsually, it is assumed that a ring has an identity, 1, under *, but books vary in this so be careful. \n\nFor fields, * is assumed to be commutative and that every element other than 0 has an inverse under *. For rings, * is not necessarily assumed to be commutative. if you drop the commutativity assumption for fields but keep the inverses, you get what is called a skew-field.\n\nExamples: The integers are a ring, but not a field since 2 (for example) doesn't have a multiplicative inverse. The rationals form a field. The collection of nxn matrices form a non-commutative ring which is not a skew-field, and the quaternions form a skew-field.
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Using the standard "construction" of the natural numbers, they ARE NOT embedded in the real numbers.  The real numbers are like a completion of a splitting field of a class of polynomials over the natural numbers. (Existance of transcendental numbers, such at pi and e, show that the real numbers themselves are not the splitting field of the set of polynomials over the naturals.  See any good textbook on abstract algebra for definitions of splitting fields, transcendental numbers, ...  )  As such, the natural numbers and real numbers are indeed disjoint sets.  HOWEVER!  Consider the function f which takes 1_N ( 1 in the natural numbers ) to 1_R ( 1 in the real numbers ), 2_N to 2_R, ... etc.  \n\nf( m_N + n_N ) = m_R + n_R\nf( m_N * n_N ) = m_R * n_R\n\nThese properties show that the map f is a homomorphism.  Moreover, the only element which maps to 0_R is 0_N showing that the map is, indeed, an embedding.\n\nWhile the sets themselves may be disjoint, the set of natural numbers is "isomorphic" to what we consider the natural numbers sitting inside the reals.  As we typically consider such sets only "up to isomorphism", we can thus assume that the natural numbers do indeed sit inside the reals.
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ok .....\n\ni think its....\nf(g(x))\n 5(x^2 - 1) + 3\n 5x^2 - 5 + 3\n =5x^2 - 2\n                        \ni think thats it....
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Use javascript to check the value of the file field. Otherwise, you can use if(is_uploeaded_file) {//some code here }
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Abstract mathematics is a huge field, with many parts. I would say, the simplest way to think about is that it is not focused on applications, like problems from engineering, or physics, or chemistry or economics. The problems in pure math, or abstract math, come from the field itself. \n\nIn real math, there is very little done with arithmetic. It all is about constructing rules for systems, called axioms, and then finding out what the implications of those axioms are. The main activity is doing proofs, which can be very challenging. \n\nIt includes things like set theory, and number theory. \n\nAnd for 2+2=5: if you set up your axioms the right way, you can prove anything. For example, what do the symbols 2, +, = and 5 mean anyway?
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First show that the reciprocal of every element of F[[x]] can be written as a Laurent series: this is best done by inductively defining the coefficients of the quotient. Conversely, show that the reciprocal of every Laurent series can be expressed as a Laurent series. This shows that F((x)) is a field (assuming you check the obvious properties) and contains the field of fractions of F[[x]]. Since a sufficiently high power of x can be multiplied by a Laurent series to get an element of F[[x]], this completes the proof that F((x))) is the field of fractions of F[[x]].\n\nAs for Z[[x]]. The idea is to show that the power series for e^x (which is clearly in Q[[x]]) is not of the form f(x)/g(x) for f,g in Z[[x]]. But if f/g is in lowest terms, it is easy to see that g must be a constant (otherwise there would be terms of the expansion with negative exponents), i.e. g is in Z. But the expression for e^x cannot be so written!